Daniel J Lips
email: d.lips@jbz.nl

Henrieke W Schutte

Ragna L A Van Der Linden

Anneriet E Dassen

Adri C Voogd

Koop Bosscha

SENTINEL LYMPH NODE BIOPSY TO DIRECT TREATMENT IN GASTRIC CANCER. A SYSTEMATIC REVIEW OF THE LITERATURE

Keywords: early gastric cancer, sentinel lymph node, biopsy

Gastric cancer is one of the main causes of cancer-related deaths around the world. The prevalence of early gastric cancer (EGC) among all gastric cancers of 45%-51% in Japan, but only 7%-28% in Western countries. The prevalence of EGC is growing partly because of better diagnostics and screening programmes. Possible treatment options for EGC treatment are expanded by the introduction of endoscopic mucosal resection and endoscopic submucosal dissection Therefore, detailed knowledge about nodal metastatic risk is warranted. We performed a systematic review of the literature concerning studies investigating the role of sentinel lymph node biopsy in EGCr and whether there is enough prove to introduce SLN as a part of treatment for EGC in the Netherlands. Several detection substances (dye or radiocolloid) and injection methods (submucosal or subserosal) are investigated. An overall sensitivity percentage of 85.4% was found. In comparison, high and clinically sufficient percentages were observed for specificity (98.2%), negative predictive value (90.7%) and accuracy (94%). Subgroup analyses showed that the combination of dye and radiocolloid detection substances is the best method for sentinel lymph node detection in early gastric cancer. However, the precise method of sentinel lymph node biopsy in EGC has to be determined further. Large, randomized series should be initiated in Europe to address this issue.

Introduction

Early gastric cancer (EGC) is defined as tumor invasion confined to the mucosa or submucosa, irrespective of nodal metastases, i.e. T1N0-2 (see Table 1). This definition reflects an appreciation that EGC represents a subset of gastric cancers that has a favorable prognosis compared to invasive gastric cancers that extend beyond the submucosa (T2-4). Gastric cancer is one of the most common causes of cancer mortality worldwide. The incidence of gastric cancer and the percentage of gastric cancerrelated deaths are unevenly distributed throughout the world. The disease is most common in the Eastern World with the prevalence of EGC among all gastric cancers 45%-51% in Japan, but only 7%-28% in western countries. [START_REF] Yuasa | Survival after surgical treatment of early gastric cancer, surgical techniques, and long-term survival[END_REF] Differences in treatment strategies for EGC between Japan and western countries are maintained. [START_REF] Yuasa | Survival after surgical treatment of early gastric cancer, surgical techniques, and long-term survival[END_REF] Most screening programmes for gastric cancer have been developed in Asian countries. [START_REF] Hamashima | The Japanese Guidelines for Gastric Cancer Screening[END_REF] In the Netherlands the primary curative therapy of all stages of gastric cancer consists of the complete resection of the tumor and prophylactic lymph node dissection. The extent of lymph node dissection is still much debated.(3) However, the need for a prophylactic lymph node dissection in EGC is probably negligible with respect to locoregional recurrence and overall survival, because of the very low risk of finding positive lymph nodes (see Table 1). [START_REF] Gotoda | Incidence of Lymph Node Metastasis From Early Gastric Cancer: Estimation With a Large Number of Cases at Two Large Centers[END_REF][START_REF] Hohenberger | Gastric Cancer[END_REF] It is estimated that 80 to 97% of the patients with EGC do not benefit from a lymphadenectomy, but are harmed by the coinciding morbidity. Non-invasive and endoscopic techniques as computed tomography (CT), magnetic resonance imaging (MRI) and endoscopic ultrasonography (EUS) are limited by their low sensitivity rates in predicting lymph node metastases in gastric cancer. For this problem sentinel lymph node biopsy could be helpful. [START_REF] Dassen | FDG-PET Has No Definite Role in Preoperative Imaging in Gastric Cancer[END_REF] Investigators from the East and West are stimulated to conduct multiple studies investigating the role of sentinel lymph node (SLN) biopsy in EGC in the past decade. The surgical treatment of early gastric cancer relates to the extend of lymphadenectomy. Therefore detailed knowledge about the nodal metastatic risk is warranted. In this systematic review we investigate whether there is enough prove to introduce SLN as a treatment for EGC in the Netherlands.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Lips DJ, et al.

SLN biopsy in gastric cancer 4

Methods

Systematic search

Relevant studies were identified through a search of the electronic database PubMed. The following search terms were used: "gastric cancer", "gastric carcinoma", "early gastric cancer", "sentinel lymph node", "sentinel node". Inclusion criteria were: study objective to investigate the role of SLN biopsy in gastric cancer, article in the English language and inclusion of at least 10 patients in the study.

Twenty-one studies fulfilled the inclusion criteria. Subsequently, reference lists of the selected studies were crosschecked. The selection and analysis of studies has been performed independently by three investigators (DJL, HWS, RLAL, AED). Studies available for analysis (ref [START_REF] Hiratsuka | Application of Sentinel Node Biopsy to Gastric Cancer Surgery[END_REF][START_REF] Kitagawa | Radio-Guided Sentinel Node Detection for Gastric Cancer[END_REF][START_REF] Hundley | Lymphatic Mapping for Gastric Adenocarcinoma[END_REF][START_REF] Ichikura | Sentinel Node Concept in Gastric Carcinoma[END_REF][START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Miwa | Mapping Sentinel Nodes in Patients With Early-Stage Gastric Carcinoma[END_REF][START_REF] Simsa | Sentinel Node Biopsy in Gastric Cancer: Preliminary Results[END_REF][START_REF] Ryu | Prediction of Lymph Nodes Metastasis by Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Tonouchi | Lymphatic Mapping and Sentinel Node Biopsy During Laparoscopic Gastrectomy for Early Cancer[END_REF][START_REF] Uenosono | Evaluation of Colloid Size for Sentinel Nodes Detection Using Radioisotope in Early Gastric Cancer[END_REF][START_REF] Kim | Lymphatic Mapping and Sentinel Node Biopsy Using 99mTc Tin Colloid in Gastric Cancer[END_REF][START_REF] Isozaki | An Assessment of the Feasibility of Sentinel Lymph Node-Guided Surgery for Gastric Cancer[END_REF][START_REF] Song | Lymphatic Mapping and Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Nimura | Infrared Ray Electronic Endoscopy Combined With Indocyanine Green Injection for Detection of Sentinel Nodes of Patients With Gastric Cancer[END_REF](21)[START_REF] Gretschel | Prediction of Gastric Cancer Lymph Node Status by Sentinel Lymph Node Biopsy and the Maruyama Computer Model[END_REF][START_REF] Lee | Comparative Study of the Subserosal Versus Submucosal Dye Injection Method for Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Park | Sentinel Node Biopsy for CT1 and CT2a Gastric Cancer[END_REF][START_REF] Mochiki | Sentinel Lymph Node Mapping With Technetium-99m Colloidal Rhenium Sulfide in Patients With Gastric Carcinoma[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF]) were subjected to a quality control using the QUADAS tool. [START_REF] Whiting | The Development of QUADAS: a Tool for the Quality Assessment of Studies of Diagnostic Accuracy Included in Systematic Reviews[END_REF] Quality was considered to be good for two studies, moderate for 12 studies and poor for four studies. The selection and analysis of the studies has been performed

independently by three investigators (DJL, HWS, AED).

Statistics

The following definitions were used:

SLN detection rate = number of patients with detected SLNs / total number of patients investigated. Negative predictive value (NPV) = number of patients with true negative SLN / (number of patients with true negative SLN + number of patients with false negative SLN).

SLN

Accuracy (ACC) = (number of patients with true positive SLN + number of patients with true negative SLN) / total number of patients with detected SLNs.

The accuracy of the various procedures for detecting sentinel lymph nodes and subgroup analysis were tested with Meta-Disc software using negative likelihood ratios, and subgroup analyses were performed according to detection method.(28) P < 0.05 was determined significant.

Results

An overview of the results from the studies included in the analysis is presented in Table 2. A total of 1314 patients were included in the 21 reviewed studies, with a mean number of 63 patients per study (range 13 -211; median 53). Tumor depth varied between T1 and T3 between the studies. All studies excluded patients with metastatic disease. Largely, the SLN detection methods used in these studies can be separated on the basis of detection substance employed into two groups, i.e. radiocolloid or dye. Four of the 21 studies investigated multiple modalities for SLN biopsy. [START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Nimura | Infrared Ray Electronic Endoscopy Combined With Indocyanine Green Injection for Detection of Sentinel Nodes of Patients With Gastric Cancer[END_REF][START_REF] Gretschel | Prediction of Gastric Cancer Lymph Node Status by Sentinel Lymph Node Biopsy and the Maruyama Computer Model[END_REF][START_REF] Mochiki | Sentinel Lymph Node Mapping With Technetium-99m Colloidal Rhenium Sulfide in Patients With Gastric Carcinoma[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF] In 16 of the 21 studies a detection dye (indocyanine green, isosulfan blue or patent blue) was used [START_REF] Hiratsuka | Application of Sentinel Node Biopsy to Gastric Cancer Surgery[END_REF][START_REF] Hundley | Lymphatic Mapping for Gastric Adenocarcinoma[END_REF][START_REF] Ichikura | Sentinel Node Concept in Gastric Carcinoma[END_REF][START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Miwa | Mapping Sentinel Nodes in Patients With Early-Stage Gastric Carcinoma[END_REF][START_REF] Simsa | Sentinel Node Biopsy in Gastric Cancer: Preliminary Results[END_REF][START_REF] Ryu | Prediction of Lymph Nodes Metastasis by Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Tonouchi | Lymphatic Mapping and Sentinel Node Biopsy During Laparoscopic Gastrectomy for Early Cancer[END_REF][START_REF] Isozaki | An Assessment of the Feasibility of Sentinel Lymph Node-Guided Surgery for Gastric Cancer[END_REF][START_REF] Song | Lymphatic Mapping and Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Nimura | Infrared Ray Electronic Endoscopy Combined With Indocyanine Green Injection for Detection of Sentinel Nodes of Patients With Gastric Cancer[END_REF][START_REF] Gretschel | Prediction of Gastric Cancer Lymph Node Status by Sentinel Lymph Node Biopsy and the Maruyama Computer Model[END_REF][START_REF] Lee | Comparative Study of the Subserosal Versus Submucosal Dye Injection Method for Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Park | Sentinel Node Biopsy for CT1 and CT2a Gastric Cancer[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF], while a radiocolloid ( 99m Tc tin colloid, 99m Tc sulphur colloid or 99m Tc colloidal rhenium sulphide) was used in nine. [START_REF] Kitagawa | Radio-Guided Sentinel Node Detection for Gastric Cancer[END_REF][START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Tonouchi | Lymphatic Mapping and Sentinel Node Biopsy During Laparoscopic Gastrectomy for Early Cancer[END_REF][START_REF] Uenosono | Evaluation of Colloid Size for Sentinel Nodes Detection Using Radioisotope in Early Gastric Cancer[END_REF][START_REF] Kim | Lymphatic Mapping and Sentinel Node Biopsy Using 99mTc Tin Colloid in Gastric Cancer[END_REF]21,[START_REF] Gretschel | Prediction of Gastric Cancer Lymph Node Status by Sentinel Lymph Node Biopsy and the Maruyama Computer Model[END_REF][START_REF] Mochiki | Sentinel Lymph Node Mapping With Technetium-99m Colloidal Rhenium Sulfide in Patients With Gastric Carcinoma[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF] Of these 21 studies one compared, but did not combine both methods. [START_REF] Gretschel | Prediction of Gastric Cancer Lymph Node Status by Sentinel Lymph Node Biopsy and the Maruyama Computer Model[END_REF] In 3 other studies the combination of a dye and radiocolloid was used to maximize SLN detection. [START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Tonouchi | Lymphatic Mapping and Sentinel Node Biopsy During Laparoscopic Gastrectomy for Early Cancer[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF] The method of application (pre-/peroperative and submucosal/serosal) is depending on the substance employed. All studies using a radiocolloid injected the detection substance peritumorally in the submucosal layer of the stomach via endoscopic route one day preoperatively. In nine out of 17 an endoscope was used in to inject the dye in the submucosal layer [START_REF] Ichikura | Sentinel Node Concept in Gastric Carcinoma[END_REF][START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Miwa | Mapping Sentinel Nodes in Patients With Early-Stage Gastric Carcinoma[END_REF][START_REF] Tonouchi | Lymphatic Mapping and Sentinel Node Biopsy During Laparoscopic Gastrectomy for Early Cancer[END_REF][START_REF] Isozaki | An Assessment of the Feasibility of Sentinel Lymph Node-Guided Surgery for Gastric Cancer[END_REF][START_REF] Nimura | Infrared Ray Electronic Endoscopy Combined With Indocyanine Green Injection for Detection of Sentinel Nodes of Patients With Gastric Cancer[END_REF][START_REF] Gretschel | Prediction of Gastric Cancer Lymph Node Status by Sentinel Lymph Node Biopsy and the Maruyama Computer Model[END_REF][START_REF] Lee | Comparative Study of the Subserosal Versus Submucosal Dye Injection Method for Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF], while 8/17 injected the detection substance peritumorally in the subserosal layer during laparotomy. [START_REF] Hiratsuka | Application of Sentinel Node Biopsy to Gastric Cancer Surgery[END_REF][START_REF] Hundley | Lymphatic Mapping for Gastric Adenocarcinoma[END_REF][START_REF] Simsa | Sentinel Node Biopsy in Gastric Cancer: Preliminary Results[END_REF][START_REF] Ryu | Prediction of Lymph Nodes Metastasis by Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Song | Lymphatic Mapping and Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Lee | Comparative Study of the Subserosal Versus Submucosal Dye Injection Method for Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Park | Sentinel Node Biopsy for CT1 and CT2a Gastric Cancer[END_REF] In one study the detection effect of indocyanine green was enhanced by using infrared ray electronic endoscopy (IREE).( 18) Another study using dye detection investigated the difference between the submucosal and subserosal techniques [START_REF] Lee | Comparative Study of the Subserosal Versus Submucosal Dye Injection Method for Sentinel Node Biopsy in Gastric Cancer[END_REF], three others used the submucosal dye application in combination with a radiocolloid. [START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Tonouchi | Lymphatic Mapping and Sentinel Node Biopsy During Laparoscopic Gastrectomy for Early Cancer[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF] The SLN detection rates varied from 66.7% to 100% (mean 94.7%; median 96.6) between the studies. The low SLN detection rate of 66.7% was observed in the study investigating the role of the size of radiocolloid-particles related to detection success. A detection rate of 66.7% was found with a particle size of 500 nm, while with the other investigated particle sizes of 100 and 50 nm detection rates of 100% were observed. The three dye and radiocolloid combination studies showed detection rates of respectively 100%, 100% and 97%. With all detection methods combined, a mean of 3.5 SLNs per patient was observed (range 1.6 -10.5; median 3). The highest number of SLNs per patient was found with the combination of indocyanine green and IREE; here, the detection rate was 98.8%. SLN positivity (SLN+) varied significantly between studies with a range of 5.2% to 73.4% (mean 28.1%; median 21.4%). No association could be observed between tumour depth and SLN+ as result of the given data.

Table 3 shows the overall results of all studies combined. The mean and median percentages for the sensitivity were 85.4% and 88.9% respectively with a wide range from 45.5% to 100%. Low sensitivity percentages of less then 90% were observed in 13/21 studies. [START_REF] Ichikura | Sentinel Node Concept in Gastric Carcinoma[END_REF][START_REF] Hayashi | Sentinel Lymph Node Mapping for Gastric Cancer Using a Dual Procedure With Dye-and Gamma Probe-Guided Techniques[END_REF][START_REF] Miwa | Mapping Sentinel Nodes in Patients With Early-Stage Gastric Carcinoma[END_REF][START_REF] Ryu | Prediction of Lymph Nodes Metastasis by Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Kim | Lymphatic Mapping and Sentinel Node Biopsy Using 99mTc Tin Colloid in Gastric Cancer[END_REF][START_REF] Nimura | Infrared Ray Electronic Endoscopy Combined With Indocyanine Green Injection for Detection of Sentinel Nodes of Patients With Gastric Cancer[END_REF](21)[START_REF] Gretschel | Prediction of Gastric Cancer Lymph Node Status by Sentinel Lymph Node Biopsy and the Maruyama Computer Model[END_REF][START_REF] Lee | Comparative Study of the Subserosal Versus Submucosal Dye Injection Method for Sentinel Node Biopsy in Gastric Cancer[END_REF][START_REF] Park | Sentinel Node Biopsy for CT1 and CT2a Gastric Cancer[END_REF][START_REF] Mochiki | Sentinel Lymph Node Mapping With Technetium-99m Colloidal Rhenium Sulfide in Patients With Gastric Carcinoma[END_REF][START_REF] Gretschel | Efficacy of Different Technical Procedures for Sentinel Lymph Node Biopsy in Gastric Cancer Staging[END_REF] In two of the 13 studies better sensitivity percentage of 90-100% was obtained by using the combination method(9,24), while one showed a similar result with IREE. [START_REF] Isozaki | An Assessment of the Feasibility of Sentinel Lymph Node-Guided Surgery for Gastric Cancer[END_REF] In general, a low sensitivity percentage was found for all detection substances (i.e. dye or radiocolloid) and application methods (i.e.

submucosal via endoscopic route or subserosal via laparotomy). A high sensitivity was found in studies using a subserosal dye technique or combination method. The mean and median negative predictive value (NPV) for all studies combined were 90.7 and 95.7%. The mean overall specificity (Spec) and positive predictive value (PPV) were 98.2 and 99% respectively. Positive SLNs proved to be falsepositive very rarely.( 9) Good results are seen for the accuracy (Acc) percentages (mean 94%; median 96.4%).

Subgroup analyses according to detection method (see Table 4) showed better discriminative power for the radiocolloid detection method then dye methods, irrespective of the application method of the dye. However, a tendency towards the best discriminative power was seen for the combination of dye and radiocolloid.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Lips DJ, et al. SLN biopsy in gastric cancer 7

Discussion

In this systematic review of the current literature not enough prove was found to introduce SLNB in the treatment protocol for EGC in the Netherlands. The overall sensitivity of <90% is just too low to justify immediate introduction of the SLN biopsy procedure in EGC in analogy of the requested sensitivity percentage of > 95% for SLN biopsy as in other solid organ malignancies. For a proper implementation of the SLN biopsy procedure in early gastric cancer several problems need to be solved. First, the amount of false-negative SLN biopsy results should be limited to increase the sensitivity of the procedure. A sensitivity of at least 90-95% should be guaranteed for safe introduction of the procedure in patient care. The lack of expertise is part of the explanation for the current low sensitivity. Several investigators report a significant increase in the sensitivity of SLN biopsy with growing experience with the procedure,(20,24) but it is not known exactly how many patients are needed to complete the learning curve.( 24) Although some investigators claim that they were able to optimize their technique with a learning curve of less than ten cases,(20) it seems reasonable to assume that the plateau-phase in the learning curve is reached when detection and accuracy rates are around 95%. Secondly the literature about EGC and SLN procedures is mainly coming from the Asian countries. Because the incidence of gastric cancer is higher in these countries and because the tumour types seem to differ, it may be hazardous to extrapolate the results from these studies to patients in Western countries. [START_REF] Dassen | Trends in incidence, treatment and survival of gastric adenocarcinoma between 1990 and 2007: A population-based study in the Netherlands[END_REF][START_REF] Yi | Pattern of lymph node metastases and its implication in radiotherapeutic clinical target volume delineation of regional lymph node in patients with gastric carcinoma[END_REF] One of the differences is the frequency of the locations of the gastric tumors related to the differences in risk factors. The most important risk factor for gastric cancer is an infection by Helicobacter pylori bacteria. This infection is primarily related to distal gastric cancer, [START_REF] Dassen | Trends in incidence, treatment and survival of gastric adenocarcinoma between 1990 and 2007: A population-based study in the Netherlands[END_REF] which is where most of the tumours of patients from the Eastern world are primarily located. This in contrast to the countries in the in the Western world where most of the gastric cancers are located in the cardia. This difference, among others, can be of importance for the ability of the SNL procedure to detect positive lymph nodes.

In the papers included in our review no clear distinction is made between early gastric cancer(T1) and locally advanced gastric cancer (T2-4). Therefore, not all results are comparable. However, we can postulate that the SLN biopsy procedure could be relevant for clinical practice in EGC to limit the extent of surgery. Lymph node metastases are present in 84% of all patients with gastric cancer (ref 30, table 1). Based on current knowledge prophylactic lymphadenectomy could be omitted in patients with a T1 tumor and SLN biopsy negative for metastasis. In these patients, the finding of a positive SLN biopsy should be followed by a lymphadenectomy as part of standard therapy. Lymphadenectomy should also be applied as a standard procedure for patients with T2 ,T3 and T4 tumours, because for patients with these stages of gastric cancer the risk of lymph node metastasis is well above 50%(table 1) and using SLN biopsy will not be of any benefit for them.

Considering all the above, the combination of dye and radiocolloid for lymph node detection is the best

M A N U S C
2.9 (0-7) 37.5% # Based on the results of group patients injected with 4 ml 1.25% indocyanine green, excluding the group injected with 8ml 0.63% indocyanine green, as the former is the preferred method to identify SLNs according to the investigators.

∧ Study investigating laparoscopic gastrectomy with lymphatic mapping.

∧∧ In the SNs of 2 of 3 patients with positive lymph nodes micrometastasis were found in frozen-sections by immunohistochemistry staining, not in the frozen-section examination by hematoxylin & eosin staining (diagnostic accuracy frozen biopsy 1 out of 3, 33%).

  positivity rate (SLN+) = number of patients with positive SLN / total number of patients with detected SLNs. Sensitivity = number of patients with true positive SLN / (number of patients with true positive SLN + number of patients with false negative SLN). Specificity = number of patients with true false SLN / (number of patients with true false SLN + number of patients with false positive SLN). Positive predictive value (PPV) = number of patients with true positive SLN / (number of patients with true positive SLN + number of patients with false positive SLN).

  M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Lips DJ, et al. SLN biopsy in gastric cancer 6

17 *

 17 and ** Results of respectively hematoxylin&eosin and immunohistochemistry staining on final pathological non-frozen examination.M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPTLips DJ, et al.SLN biopsy in gastric cancer 18

TABLE 1 . T-status and Observed Metastatic Lymph Node Involvement According to Depth of Tumor Invasion T- status Depth of tumor invasion of gastric wall Metastatic lymph node involvement Reference

 1 The precise role of sentinel lymph node biopsy in ECG remains to be determined. Large, randomized series should be initiated in Europe to address this issue. Advances in laparoscopic and endoscopic surgical procedures and a growing older patient population with co-morbidities require such initiatives as less-extended procedures are performed. It is recommended to perform a minimum of

	R I P T

Derived from International and Japanese Gastric Cancer Associations (Gastric Cancer 1998;1:10-24) and the International Union Against Cancer (UICC) TNM staging system (6 th edn, 2002) Tis (carcinoma in situ). Early gastric cancer is defined as tumor invasion confined to the mucosa or submucosa, irrespective of nodal metastases, i.e. T1N0. The incidence of lymph node metastasis is low in early gastric cancer patients, with more than a 90% 5-year survival rate. The overall prognosis for patients with gastric cancer remains poor, with a 5-

TABLE 2 . Results Sentinel Lymph Node (SLN) Procedures in Gastric Cancer

 2

	Reference	Method	TNM ∧ ∧ ∧ ∧	N	Detection	SLN/pt	SLN+ ¶ Sens Spec PPV NPV	Acc	Acc	Skip	SLN
					succes-	(range)						SLN	frozen	metastasis †	micrometastasis ‡
					rate							status	biopsy	(detected)
	Hiratsuka M, et al.	Indocyanine green;	T1-2
	Surgery 2001	subserosal; laparotomy		74	98.6%	2.6	12.3%	90%	100%	100%	98.4%	98.6%	-	-	-
						(1-9)
	Kitagawa Y, et al.	99m Tc tin colloid;	T1-2
	Br J Surg 2002	submucosal;		145	95%	3.6	7.8%	91.7%	100%	100%	98.4%	98.6%	-	0.8%	-
		endoscopically				(1-8)								(100%)
	Hundley JC, et al.	Isosulfan
	Am Surg 2002.

  Skip metastases are defined as lymph node metastases in D2 or D3 compartments, without involvement of the perigastric D1 compartment. Percentage calculated by the equation: patient with skip metastasis/total patients studied. ‡ SLN micrometastasis detection rate is defined by the equation: SLN micrometastasis positive/total SLN metastasis positive.

						ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT
	Lips DJ, et al. Lips DJ, et al.							SLN biopsy in gastric cancer SLN biopsy in gastric cancer
	Reference Reference	Method Method	TNM ∧ N TNM ∧ N	Detection Detection	SLN/pt SLN/pt	SLN+ ¶ Sens Spec PPV NPV SLN+ ¶ Sens Spec PPV NPV	Acc Acc	Acc Acc	Skip Skip	SLN SLN
					succes-rate succes-rate	(range) (range)						SLN SLN	frozen frozen	metastasis † metastasis †	micrometastasis ‡ micrometastasis ‡
	Nimura H, et al. Br J Surg 2004 Zulfikaroglu B, et al. Surgery 2005 Gretschel S, et al. EJSO 2005. Mochiki E, et al. Am J Surg 2006 Gretschel S, et al. Ann Surg Oncol 2007 EJSO 2010l Cozzaglio L et a.	Indocyanine green; submucosal; endoscopically + IREE 148 MBq 99m Tc filtered sulfur colloid; submucosal; endoscopically 99m Tc colloidal rhenium sulphide; + Patent blue; submucosal; endoscopically 99m Tc colloidal rhenium sulphide; submucosal; endoscopically 99m Tc colloidal rhenium sulphide; + Patent blue; submucosal; endoscopically (cb) Patent blue dye technique	T1-2 T1-3 T1-3 T1-3 T1-3 T1-2	84 32 15 19 59 35 35 35 29	98.8% 98.8% 97% 93.3% 100% 96.6% 97% 97% 97% 96,5%	5.5 10.5 2.4 (1-8) 3 (1-5) 3 3.8 (1-10) 3 (1-10) 3 (1-6) 3 (1-10) (1-3) 1.6	7.7% 8.4% 13.3% 18% 57.1% 73.4% 35.1% 64.7% 47.1% 64.7% 61%	100% 100% M A N U S C R I P T 100% 100% 100% 100% 100% 100% 100% 100% status biopsy 100% 100% 63.6% 100% 100% 100% 100% 100% 94.7% 100% 95.2% 100% --100% 95.5% 90% 100% 96.7% 97% 88.9% 100% 100% 100% 100% 100% 83.3% 100% 92.9% 100% --status biopsy 83.3% 100% 100% 89.2% 93% -91.7% 66.7% 91.7% 100% 100% 100% 100% 100% 100% 83.3% 55.6% 83.3% 94.1% 76.5% 94.1% ---A N U S C R I P T M 75% 75% 88% 55% 75% -	25% 0% (detected) --10% (100%) 11.1% (0%) 7.1% (detected) -----	----0% -18%
	Kim M-C, et al. Ann Surg 2004 Isozaki H, et al. Gastric Cancer 2004 Song X, et al. Am J Surg 2004 Lee JH, et al. EJSO 2005 Park DJ, et al. EJSO 2006	99m Tc tin colloid; submucosal; endoscopically Isosulfan blue; submucosal; endoscopically Isosulfan blue; subserosal; laparotomy (cb) Isosulfan blue; subserosal (laparotomy) versus submucosal (endoscopically) Indocyanine green; subserosal; laparotomy	T1-3 T1-3 T1-3 T1-2 T1-2	18.2% 18.8% 25.6% -20% A C C E P T E D 100% 100% 93.5% 2 (1-8) 84.6% 144 46 97.2% 3.3 (1-9) -27 96.3% 2.7 100% 34 (1-6) 71 50 91.5% 94% 2.5 2.9 16.9% 10.6% 61.1% 45.5% 100 50 94% 4.4 11.7% * 22.2% ** 78.6% * 71.4% ** A C C E P T E D	100% 100% 100% -100% 100% 100% 100% * 100% **	100% 100% 100% -100% 100% 100% 100% * 100% **	100% 100% 93.8% -100% 87% 85.7% 96.4% * 88.6% **	100% 100% 95.3% -100% 89.2% 87.2% 96.8% * 91.1% **	100% 100% 97.7% ------	9.1% 9.4% 7% (100%) -0 (0%) ----	--0% --21.7% --4.3%
						(1-6)
												15 16

Abbreviations: Acc accuracy; cb combination; IREE infrared ray electronic endoscopy; NPV negative predictive value; PPV positive predictive value; Sens sensitivity; Spec specificity; SLN sentinel lymph node.

∧ ∧ ∧ ∧

The actually observed T-status is mentioned; all studies excluded patients with metastatic disease. ¶ SLN positivity (+) for metastasis is calculated by the equation: patient with SLN positive for tumor/total patients with SLNs found. †

TABLE 3 . Summary of Results of Sentinel Lymph Node Biospy Procedures

 3 : Acc accuracy; NPV negative predictive value; PPV positive predictive value; Sens sensitivity; Spec specificity. All number are presented in percentages.

		Sens	Spec	PPV	NPV	Acc
	Mean	85.5%	98.2%	99%	90.7%	94%
	Median	88.9%	100%	100%	95.7%	96.4%
	Range 45.5-100% 75 -100% 88 -100% 50 -100% 70 -100%

Abbreviations

TABLE 4 . Negative Likelihood Ratio for Sentinel Lymph Node Biopsy Procedures

 4

		N Neg LHR	95% CI
	total	31	0,230	0,178	0,297
	dye total	17	0,283	0,209	0,382
	SM	9	0,264	0,162	0,430
	SS	8	0,308	0,2207 0,430
	Radiocolloid	11	0,163	0,108	0,247
	dye and radiocolloid	3	0,101	0,037	0,342

Abbreviations: N number of studies; neg LHR negative likelihood ratio; CI confidence interval

The included number SLN biopsy procedures (N) is 31, because all individual arms of the studies are counted.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Lips DJ, et al. SLN biopsy in gastric cancer


Frédéric Morlot
email: frederic.morlot@polytechnique.org

Franc ¸ois Baccelli
email: francois.baccelli@ens.fr

Salah Eddine Elayoubi
email: salaheddine.elayoubi@orange.com

An Interaction-Based Mobility Model for Dynamic Hot Spot Analysis

In this paper, we analyze phenomena related to user clumps and hot spots occurring in mobile networks at the occasion of large urban mass gatherings in large cities. Our analysis is based on observations made on mobility traces of GSM users in several large cities. Classical mobility models, such as the random waypoint, do not allow one to represent the observed dynamics of clumps in a proper manner. This motivates the introduction and the mathematical analysis of a new interactionbased mobility model, which is the main contribution of the present paper. This model is shown to allow one to describe the dynamics of clumps and in particular to predict key phenomena such as the building of hot spots and the scattering between hot spots, which play a key role in the engineering of wireless networks during such events. We show how to obtain the main parameters of this model from simple communication activity measurements and we illustrate this calibration process on real cases.

I. INTRODUCTION

Traces collected by wireless operators in urban environments during multi event mass gatherings reveal that user mobility is extremely volatile and leads to the formation of hot spots, exemplified on the snapshot of Figure 1 which was gathered during one the 2008 Paris Fête de la Musique, where unpredicted gatherings are triggered by a large number of spontaneous street concerts taking place throughout the city. Such observations have also been made during other events, like a summer festival in Poland and Euro2008 soccer matches in Spain and in Romania. A key phenomenon within this context concerns the dynamics of these hot spots, namely their formation and their disaggregation which is random both in space and time.

Accurate and yet tractable user mobility models allowing one to represent these phenomena would significantly improve the robustness and the realism of the simulators and the analytical models used by wireless communication operators. They would hence ameliorate the understanding of the impact of mass mobility on QoS within this setting. Predicting hot spots dynamics could also help in the design and the tuning of adaptive radio resource allocation schemes, where it is fundamental to have an idea of some time caracteristics of the system. This allows a better utilization of the corresponding infrastructure and eventually a better QoS and better services to end users. As existing models are not completely adapted to the phenomena we observe on traces, such as the random dynamic hot spot phenomena alluded to above, we propose a new mobility model taking into account interaction between different users and show by a mathematical analysis based on Markov chain theory that this interaction-mobility model is sufficient to generate random hot spots. For this, we will represent an urban area as a collection of potential hot spot zones. These zones represent potentially attractive points in the city during such mass gatherings.

The basic assumption in our model is that users influence each other. In Section III, we first study the "sheep model", where this mutual influence is particularly strong and where hot spots form at random in time and space, but are stable once formed (i.e. there is no disaggregation of hot spots in this model yet). We start with the 2-zone case, in which we find closed forms for the mean time for the formation of a stable hot spot and for the law of its location. We also give asymptotic estimates for these quantities, and upper bounds on the convergence speed. We then generalize the previous results to an arbitrary number of zones.

Section IV introduces the "sheep and maverick" model, which is more realistic, yet tractable, and which features all key phenomena identified on the traces and alluded to above : the formation of hot spots at random times and places, that we will refer to as filling, their disaggregation that we will refer to as scattering, and the dynamics between them.

In a last part, we leverage filling and scattering in the "sheep and maverick" model to calibrate our mobility model and show that it can be made consistent with the traces.

II. THE NEED FOR A NEW MODEL

First, let us describe more precisely the phenomenon we want to model. A first remark is the existence of hot spots, that follow some dynamics. On the snaphots of Fig. 2, the crowd moves in large numbers outside a zone : we shall call that a hot spot scattering. A new or a few new hot spots then build up at other locations : we shall call that a hot spot filling.

We now survey existing mobility models and discuss how they could be adapted to cope with these phenomena.

The most popular mobility model used in communication networks is probably the Random Waypoint Model (see [START_REF] Boudec | Perfect Simulation and Stationarity of a Class of Mobility Models[END_REF]). This model is easy to simulate and is analytically tractable in many situations depending of the mobility rules of the individuals on the domain (wrap around, edge reflection...). However, it applies to a single mobile user, or to a collection of users moving independently.

Using this model, in order to take clumping into account, we would have to choose a trip selection rule which gives more mass to potentially attractive zones. As a consequence, the law of the location of a single user would clearly favour these zones. However, such a situation prevents any macroscopic time dynamics. Indeed, with a large number of independent users, all following the same law, each snapshot would be more or less the same, very close to the theoretical location law 1 .

Thus, all clumps would happen at predefined locations and the sudden hot spot scattering as well as the sudden new hot spot aggregation observed in our traces would happen with extremely small probability.

More generally, any multi user model built from the superposition of many independent single user motions will exhibit the same "lack of hot spot dynamics", while a key point observed within traces is a strong dependence between users, since there is clearly a joint motion from a hotspot to another leading to reinforced clumping in the corresponding zone.

Nomadic Community Mobility Model or Reference Point Group Mobility Model (see [START_REF] Camp | A Survey of Mobility Models for Ad Hoc Network Research[END_REF]) consist in making small groups of users move in a mean given direction, while the users themselves are allowed to walk here and there around the imposed direction. These models are well adapted for a group of visitors in a museum for instance, where the trajectory is more or less the same for everyone. However, such a system does not allow us to make groups scatter.

Let us now review some ideas about modelling users' interactions. Since we need a strong dependence among users, 1 This follows directly from Sanov's theorem (see [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], Theorem 6.2.10). Let us denote by µ the law of the location of a single user. Let X 1 , X 2 , . . . Xn be the locations of n users randomly and independently located according to µ. Let μn = 1 n n k=1 δ X k be the empirical measure associated with a snapshot : Then for all given events B, Sanov's theorem states that, for large values of n, P(μn ∈ B) ∼ exp [-n inf ν∈B H(ν|µ)], where H(ν|µ) denotes the Kullback-Leibler divergence between the two probability measures µ and ν. In particular, if B represents the fact that users do not clump where they should according to µ, H(ν|µ) will be large for ν ∈ B, and the probability to make such an observation will be extremely small when n is large. a natural attempt could be to work with permanental processes ( [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF]) that are known to exhibit frequent clumping. However, permanental processes are nothing but particular Cox processes (see [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF], Remark 3.6.3.). This means that we would have to choose an underlying intensity measure which varies over time to cope with the dynamics of clumping zones, which is as difficult as building directly a convenient model.

As for determinantal processes, very little has been written yet about time-dependent versions. One way consists in starting from a gaussian analytic function, for instance f (z) = n≥0 a n z n , where the a n 's are i.i.d. Gaussian. Such a series converges almost surely on the unit disk, and its zeros form a determinantal process. It is then easy to make the a n 's time-dependent, by using Ornstein -Uhlenbeck processes for instance (see [START_REF] Uhlenbeck | On the theory of the Brownian Motion[END_REF]). The spatial processes we obtain are in fact almost exclusively distributed on the edge of the unit disk 2 . Such a phenomenon is too extreme for our purpose. For instance, it would be better adapted to model a wave of panick.

Another class of mobility models that could be considered is the class of spatial birth and death processes (see [START_REF] Garcia | Spatial birth and death processes as solutions of stochastic equations[END_REF]) first studied by Preston in 1975 (see [START_REF] Preston | Spatial birth-and-death processes[END_REF]). Such a process can simulate characteristic point patterns if one chooses its 2 In fact, this is a direct consequence of the Peres & Virág theorem (see [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF], Theorem 4.1.1.), which states that the n-th joint intensities of the zeros is given by :

ρn(z 1 , . . . , zn) = 1 π n det 1 (1 -z i zj ) 2 . In particular, ρ 1 (z) = 1/π(1 -|z| 2 ).
stationary distribution in a proper manner. Furthermore, in contrast to the random waypoint model, it provides a real time dynamics. However, this model is known to lead to Gibbs point processes which are usually not analytically tractable. This state of the art leads us to the conclusion that there is a need for a tractable model allowing one to exhibit the type of hot spot dynamics observed in communication traces.

III. THE SHEEP MODEL

The aim of this section is to model user clumping in some predefined zones. The basic idea is that users tend to move towards places where there are already more people, creating a self-enhanced clumping.

A. General Overview

Let us divide the city into K zones, that are exchanging mobiles one with another. In each zone, a mobile waits during an exponential time with parameter µ before deciding whether to change its zone or not.

In a first step, let us suppose that the mobiles are totally influenced by their environment : they behave as sheep. They decide to move towards one zone or another with a probability proportional to the current repartition of the population (see Fig. 3) of the zone. In other words, if we denote by N the total number of mobiles, by n 1 the number of mobiles in the first zone, by n 2 the number of mobiles in the second zone, etc., a mobile from the k-th zone decides to move to the k ′ -th zone with probability n k ′ /N . Conversely, he decides to stay in the k-th zone with probability :

n k N = 1 - k ′ =k n k ′ N .
Obviously, there are K absorbing states, which correspond to the situation where all mobiles are in the same zone (these are the K stable hot spots mentioned in the introduction). If we denote a state by n = (n 1 , . . . , n K ), then an absorbing state is given by : N e k = (0, . . . , N, . . . , 0)

k-th position , 1 ≤ k ≤ K.
We first study the 2-zone case for the sake of progressive exposition.

B. The 2-zone Problem

In this section we take K = 2. Let us denote by n(t) = (m(t), N -m(t)) the state of the system at time t. Each mobile stays in his zone during a random exponential time with parameter µ, before deciding whether to move or not. All waiting times are supposed to be independent. In other words, the system features a superposition of N exponential clocks (m clocks in the left zone and N -m in the right zone).

In the left zone for instance, the superposition of all waiting times should be an exponential clock with parameter µm, but there is not always a transition, since mobiles have the choice between moving and staying. Since the probability of moving is (N -m)/N , the actual resulting clock is a thinning of the µm-clock with probability (N -m)/N . As a conclusion, it is a µm(N -m)/N -clock. The same proof could be derived for the right zone.

Thus, the whole process is Markov, with transitions described in Fig. 4. Let us denote by Q = (q m,m ′ ) the infinitesimal generator. 1) Mean Absorption Time: We are interested in characterizing mobility between different zones. For instance, we would like to give an estimate of τ , the first time to reach an absorbing state, starting from the current state of the network n : τ = inf{t/n(t) = (0, N ) or (N, 0)}. The average value of τ is given by h n = E[τ |n(0) = n]. We know (see [START_REF] Norris | Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics[END_REF], Theorem 3.3.3. about hitting times) that the vector h is the minimum solution of the equation :

h (0,N ) = h (N,0) = 0 ∀m / ∈ {0, N }, 0<i<N q mi h (i,N -i) = -1.
(

) 1
We prove in appendix A that :

h m,N -m = 1 µ   0≤i<m N -m N -i + 0≤i<N -m m N -i   . (2)
2) Asymptotic Estimate: Typically, N corresponds to the number of users in a city district, thus it is quite large. In order to speed up computations, it can be useful to use an asymptotic estimate of h n . Let x = n 1 /N , for 1 ≤ n 1 ≤ N -1. We want an estimate that can be used for large values of N , and for all values of x. This means that the quality of this estimate should only depend on N . Since N is large, we consider x as a continuous variable taking its values in ]0, 1[. The computation in appendix A gives :

h n ≈ - N µ ((1 -x) log(1 -x) + x log(x)) (3)
and r n ≤ 2/µ, where r n is the absolute error of the approximate. Note that (3) is always an equivalent in the mathematical sense, even when h n is minimal, that is, when n = 1/N or (n -1)/N . Indeed, in that case the approximate is equal to :

- N µ 1 N log 1 N + N -1 N log N -1 N ∼ log(N ) µ ,
whereas r n remains bounded. Fig. 5 exemplifies the approximation curve. 3) Absorption Probability: So far we have obtained a closed form for the mean absorption time, as well as an asymptotic estimate. Now we would like to determine the probability to reach state (0, N ) rather than state (N, 0), starting from n = (m, N -m). We shall denote this probability by p n . Using Q, we obtain the transitions for the imbedded Markov chain of the process (see Fig. 6). Conditioning with respect to T 1 , the first transition time, and using Markov property, we get

p n = 1 2 p n+(-1,1) + 1 2 p n+(1,-1) .
Using the fact that p (0,N ) = 1 and p (N,0) = 0, we conclude immediately that :

p n = n 2 /N. (4)

C. More Clumping Zones

We now consider the same problem with K zones. 1) Description of the system: Now a state is a K-uple n = (n 1 , n 2 , . . . , n K ), with n k = N . Hence the space of possible states is a (K -1)-dimensional variety. A transition from the k-th zone to the k ′ -th zone is given by :

n → n ′ = n + e k ′ -e k (5)
(we merely introduced a migration operator in Kelly's sense, see [START_REF] Kelly | Reversibility and stochastic networks[END_REF], 2.3.). In the most general case, there are K(K -1) possible transitions (corresponding to existing couples (k, k ′ )).

The transition rate corresponding to the mobility k → k ′ is given by : 2) Absorption Time: Generalizing equation (2) leads to :

q nn ′ = µn k n k ′ N .
h n = 1 µ 1≤k≤K 0≤i<n k N -n k N -i (6)
(for a proof, see appendix B). On fig. 7, we propose a 2-dimensional surface representing the mean absorption time for K = 3.

Then, defining x k = n k /N , it is immediate to generalize equation ( 3) to find :

h(x) ≈ - N µ k (1 -x k ) log(1 -x k ) (7)
and |r(x)| ≤ K/µ.

3) Absorption Probability: In the same way as in section III-B3, we can compute the probability to reach an absorbing state rather than another one, for example say N e K . We use recursive formulas between a state and its neighbours. For example, in the most general case, we have :

p n = = (k,k ′ ) n k n k ′ p n+e k ′ -e k = (k,k ′ ) n k n k ′ .
Using then the fact that ∀k = K, p N e k = 0, we obtain all the p n 's recursively. In fact, it is quite easy to generalize equation ( 4) to obtain p n (K) = n K /N = x K .

IV. MAVERICKS AND HOT SPOTS DYNAMICS

So far, we have modeled the phenomenon of clumping in one given zone, which is only an aspect of what can be observed in dense urban networks during exceptional events. The sheep model has a major drawback, which prevents it to fit reality : as soon as it has reached an absorbing state, it does not evolve any more. This does not account for scattering, where a hot spot suddenly disaggregates before reforming in other zones.

This section aims at modeling scattering. We introduce some users, the mavericks, who are not influenced by the others. In the model, mavericks coexist with sheep. During a transition, mavericks choose uniformly their target zone, including their current zone, independently from other mobiles (which is a simple instance of independent random waypoint motion). In this new model, states of the form N e k are not absorbing anymore.

We introduce maverickness as follows 3 : at each transition, an individual decides to behave as a maverick with probability α (α is the maverickness rate), and as a sheep with probability 1 -α. The new rate of the (k → k ′ ) transition is hence given by :

q nn ′ = µn k α • 1 K + (1 -α) n k ′ N .

A. Mean Reaching Time

We call reaching time the time to reach any extremal state, namely a state of the form N e k from some k. On Fig. 9, we have plotted the evolution of the mean reaching time with respect to α.

As α grows, one can observe several phenomena : 3 There are several possible coexistence models. A first model would consist in choosing a fixed population of sheep and mavericks at the beginning of the evolution; the main objection against this model is that it is quite constrained : suppose there are only 20 individuals; if we want to reach a "maverickness" rate of 0.01% for example, this is impossible. • The mean reaching time grows exponentially with α. On Fig. 10, we have plotted the maximal mean reaching time with respect to α (this corresponds to the case where all zones initially contain each approximately N/K individuals). • On Fig. 9, as α increases (in other words, as the system becomes saturated by mavericks), the surface becomes very quickly plateau-shaped (this is already the case for α = 0.08). Apart from the regions close to extremal states, all reaching times are more or less the same, which means that reaching times become insensitive to initial state. Even if the system starts from a state close to an extremal point, it takes a large excursion and a long time to reach one of the extreme states. It is interesting to observe that such a small proportion of mavericks can induce such a perturbation (for further details, see section IV-B). This is not yet a proof but just an indication that there is scattering (we propose a systematic study of scattering in section IV-C).

B. Phase Transition

To characterize the phase transition observed in Fig. 9, we calculate the invariant measure π of the "sheep and maverick" Markov process, and observe whether there is a value of α for which all configurations are equiprobable. π is given by the equation πQ = 0. Let β = α K(1-α) , we have :

π(n) = A K k=1 1 n k ! n k -1 i=0 (N β + i), ( 8
)
where A is a normalizing constant. This result is proved in appendix C. On Fig. 11, we plot the invariant measure with respect to α for K = 2 and N = 50. As already indicated by Fig. 9, it appears that when α grows, the influence of mavericks is more and more noticeable. For higher values of α, the probability is more concentrated on uniformly-spread states. There is a critical value α c where the curve's shape reverses. What does the transition look like? Is it possible to obtain a completely flat curve (or surface in the general case) ?

Expression [START_REF] Kelly | Reversibility and stochastic networks[END_REF] shows that it is possible. It is sufficient to impose N β -1 = 0 to obtain a uniform distribution. This corresponds to the critical value :

α c = K N + K .
• for α < α c , the influence of the sheep remains noticeable. This is the most interesting case, because the system still exhibits clumping.

• for α = α c , all states are equivalent4 .

• for α > α c , the sheep loose very fast their influence. As traces reveal the existence of clumping, we shall assume until the end of the paper that α < α c .

C. Macro-states and Hot Spot Dynamics

The system is fully characterized by four parameters, which are N , K, µ and α. The three first parameters can be directly measured, but α cannot. And yet, it is of major importance. In a sense, it represents the propensity of users to clump. How can we make it best fit reality ? We would like to find a typical characteristic of the system, easy to measure and strongly related to α. This is what this subsection aims at.

Let us study further the case α < α c . If we simulate such a system, we observe clumping as expected, but we also observe scattering. We would like to evaluate how often a scattering happens. However, extremal states of the form N e k are extremely rare, so that it would be difficult to base our analysis on them. We will rather consider "macro-states", which are the union of several states close to one of the extremal states.

For example, let us denote by A k the macro-state in which the k-th zone contains a significant proportion of the population : A k = {n : n k > N sup }, where N sup is an lower bound for a zone to be considered as dense. Conversely, we define B k = {n : n k < N inf }, where N inf is a lower bound. Then a scattering is simply a transition from A k to B k . More precisely, for k fixed, we define the corresponding scattering time S k as the duration between the first instant when we reach A k and the first instant when we reach B k after that. Another option would have consisted in using the last time we leave A k before reaching B k in place. But this is not a stopping time, which would make such a choice more difficult to handle using Markov chain theory. Hence the former definition.

The expectation of S k only depends on initial state, regarding that we start in A k or not. If we do not start in A k , then almost surely n k will be equal to N sup + 1 when we reach A.

It is shown in appendix D that E[S|n (0) = n] is given by the following analytic expression :

N µ(1 -α) n k ∨(Nsup+1) l=N inf N j=l l≤i<j δ(i) j(N (K -1)β + N -j) , (9)
with

δ(i) = (N -i)(N β + i) i(N (K -1)β + N -i) .
The expression in (9) only depends on the initial state, and in particular on whether it belongs to A k or not. If we are not in In the same way, we can compute the mean "filling time" F of a zone, which is the dual of the scattering time, and which we propose to define as the time for the population in a given zone to grow from N inf to N sup . We have :

E[F ] = N µ(1 -α) N -N inf +1 l=N -Nsup N j=l l≤i<j ǫ(i) j(N β + N -j) , (10)
with

ǫ(i) = 1 δ(N -i) = (N -i)(N (K -1)β + i) i(N β + N -i) .
Note that it is easy to find some recursive formulas for the computation of E[S] and E[F ], which allows to decrease the complexity of their evaluation down to a Big-O of N .

V. CALIBRATION WITH TRACES

The aim of this section is to see whether the crowd movements observed in the traces can be described by our "sheep and maverick" mobility model.

We propose to use E[S] or E[F ] for our calibration. These quantities can be measured. Let us now see that if E[S] and N are known for instance, we can then deduce α by solving an inverse problem. If the function E[S] = f (α) is injective, at least for α < α c , this inverse problem can be solved without ambiguity. It appears that f (α) is always convex, thus decreasing on ]0, α min [, so that a necessary and sufficient condition would be α c ≤ α min , provided we prove convexity. On Fig. 12 this is always the case, but this is not true in general; a counter-example is K = 2, N = 100, N sup /N = 0.99%. In practice, α min seems to be hard to compute analytically, so that we could not find a simple sufficient condition.

Our experiences show that such counter-examples appear to be extremely rare, especially when K > 2. It seems that most choices of N inf and N sup satisfy the injectivity of f (α) on ]0, α c [. However, if it turned out that it is impossible to choose N inf and N sup properly, we advise to complement the measure of E[S] with a measure of E[F ] for instance.

In contrast, when measuring just E[F ], one has to be more cautious, since the injectivity of g(α) = E[F ] on ]0, α c [ seems less common.

Let us consider for instance the transfer that happens between Parc des Princes and Auteuil (see Fig. 2). Observing carefully our data, we conclude that there are three attracting zones :

• Auteuil itself • the Parc des Princes, northern part • the Parc des Princes, southern part so that we take K = 3. On Fig. 13, one can see the numbers of users in each zone. The total number of traces remains more or less constant, so that we choose an average value of N = 4830. This gives α c = 6.2 × 10 -4 . In our data, a SMS signalization lasts about one second : each time a SMS is sent or received, a new user is counted, but he is deleted after one second. Therefore, even if this is not true, but in order to remain consistent, we make as if users were staying one second in the zone before disappearing. Thus we put µ = 1 sec -1 . Choosing to calibrate first on Auteuil filling, we take N sup /N = 55% and N inf /N = 45%. After verifying that g(α) is injective on ]0, α c [, we find α = 0.6 × 10 -4 , that is, α ≪ α c . This was to expect, since scattering is very clear in our case.

In order to verify this value, we also measure the Parc des Princes South scattering, which yields α = 0.5 × 10 -4 . Now, if our model is consistent, it has to predict other clumps or scattering over Paris. Let us consider the triangle Châtelet-Saint Michel-Bastille for instance, in the heart of Paris (see Fig. 14). These zones are attractive. Besides, we consider a fourth zone which is the complementary region inside the triangle. That one experiences scattering, whereas the three attractive zones experience filling at the same time. Theoretically, a filling of a zone from 5% to 28% for instance should last about 10800 seconds, according to the value of α we found previously. Now, the filling of Bastille lasts about 11300 seconds, so does the filling of Châtelet, and the filling of Saint Michel lasts about 11000 seconds, which is very close to our prediction.

VI. CONCLUSION AND PERSPECTIVES

In this paper, we proposed a new mobility model able to represent and quantify the hot spots found on traces, along with their random time dynamics. Our aim was not to build a generic mobility model, adapted to every kind of situation, but rather to focus on multi-event mass gatherings. We have proved our model to be analytically tractable, by derivating closed forms and asymptotic estimates. Using measurements from a GSM network, we also showed how to calibrate the model to fit reality. In addition, our model was proved to be able to predict the order of magnitude of hot spots in different time-space situations.

Apart from the main application, which consists in building an efficient mobility model for network simulators, we describe here some applications, that rely on the ability to predicting clumping or scattering with a good likelihood.

A. Dynamic Spectrum Allocation and Software Defined Radio

Dynamic Spectrum Allocation (see [START_REF] Leaves | Dynamic Spectrum Allocation in Composite Reconfigurable Wireless Networks[END_REF]), consists in allocating variable bandwidth resources, according to differrent zones and different densities of population. As for Software Defined Radio (see [START_REF] Mitola | Cognitive Radio : an Integrated Agent Architecture for Software Defined Radio[END_REF]), it consists in deploying base stations that are able to switch from a system (like GSM) to another one (like UMTS), depending on the demand.

Even if they have not been implemented in real networks yet, these two domains are today quite an active field of research. It is crucial that implemented algorithms can have an idea of the typical evolution durations of the system.

For instance, if we can predict if a zone that is full will remain so for a long time, or if an empty zone will soon fill, one can decide with much more accuracy whether allocating resources or not.

More generally, some optimization algorithms are known to depend massively on the tuning of a key parameter. For instance, [START_REF] Kauffmann | Measurement-Based Self Organization of Interfering 802.11 Wireless Access Networks[END_REF] proposes an algorithm to optimize radio ressources in a 802.11 network, based on Gibbs sampler. The "temperature parameter" of the sampler has to be tuned very carefully, so that the optimizer can react quicklier than the typical time of evolution of the system itself. More concretely, in our case, supposing that we want to take users mobility into account in a 802.11 network, if we can anticipate a scattering, we are able to adapt the optimizer in a suitable manner.

Also in the case of a genetic algorithm, it can be interesting to adapt the cross-over and the mutation rates of the optimizer to react to a sudden evolution of the system. Indeed, the higher these rates, the stronger the diversity of the genetic population. In our case, a scattering is likely to modify drastically the system to optimize. Thus it is useful broaden preventatively the variety of solutions that are explored.

B. Data Caching in Dense Ad-hoc Networks

In a dense data ad-hoc network, it can be useful to add some fixed servers which keep the most frequently asked data in cache memories. Various algorithms already exist on this subject (see for instance [5][15]). The question is where and when should we cache memory ?

The idea is to select some strategic places where people usually clump together. These are our K zones. Suppose that, using section V, we have previously evaluated α over the study area, possibly for a different number of users and zones. Then, measuring in real-time the actual number N of users and computing the corresponding scattering times, as soon as a zone exhibits clumping, one knows how long on average this will last. In other words, one knows if it is worth caching data at that place.

Conversely, if a zone is almost empty, one knows, using "filling times", if there is a risk that we will soon encounter a peak of population and have to rush to cache memory there. transition probabilities). If there exists a transition n → n ′ with probability > 0, then p nn ′ > 0, and p nn ′ = 0 otherwise.

Since h n ′ = 0 if n ′ is absorbing, the second line of (11) can be rewritten as :

n ′ q nn ′ h n ′ = 0 ⇔ n ′ =n n→n ′ exists p nn ′ h n ′ = h n .
Consider n such that h n is maximal. Then by convexity, since all coefficients p nn ′ are nonnegative, we see that necessarily all h n ′ s such that n → n ′ exists with probability > 0 are equal to h n . Step by step, since we can reach at least one absorbing state starting from n, this proves h n = 0.

Finally, changing h in -h and applying the same result, we see that the minimum of the h n s is zero too, which achieves the proof.

C. Steady-state Computation

In this section, we will explain in several steps how to prove [START_REF] Kelly | Reversibility and stochastic networks[END_REF]. First let us take K = 2 and keep notations of section III-B.

We are facing with a birth and death process, which is timereversible, so that we can directly try to satisfy local balance equations : ∀m ∈ 0, N -1 , π(m)q m,m+1 = π(m + 1)q m+1,m .

We deduce at once :

π(m) = π(0) m i=1 (N -i + 1)(N β + i -1) i(N β + N -i) .
In order to generalize this formula, we would like to rewrite it in a symmetric manner w.r.t. n 1 and n 2 . In fact, we have :

π(m) = π(0) n1 i=1 N β+i-1 i n2 i=1 N β+i-1 i N i=1 N β+i-1 i
. This is equation [START_REF] Kelly | Reversibility and stochastic networks[END_REF] in case K = 2. Now we can reasonably suppose that (8) still holds in the general case. In fact, this is quite immediate. Indeed, thanks to the product form of (8), we know by the same calculation that in the general case, the partial balance equations are still satisfied. This completes the proof. [START_REF] Boudec | Perfect Simulation and Stationarity of a Class of Mobility Models[END_REF]

D. Proof of equation

Fig. 1 .

 1 Fig. 1. Communication activity during Fête de la Musique in Paris, on June 21 st , 2008. It is 11 :23 pm. One can observe important hot spots in the center (Châtelet, Bastille, Saint-Michel) as well as at the Parc des Princes (the city stadium) and Auteuil (the city park).

Fig. 2 .

 2 Fig. 2. Communication activity in Paris on June 21st, 2008, between 07:47 pm and 11:17 pm. One can observe scattering from the Parc des Princes (south) to Auteuil (north).

Fig. 3 .

 3 Fig. 3. The sheep model. Here, N = 7 and K = 4. The mobile in the upperleft corner has 4 choices. The four decision probabilities are proportional to the numbers of mobiles.

Fig. 4 .

 4 Fig. 4. Diagram of the markovian transitions.

Fig. 5 .

 5 Fig. 5. Asymptotic estimate of hn [in min]. N = 80 and µ = 1 min -1 . The circles represent hn, and the continuous line represents the estimate.

Fig. 6 .

 6 Fig. 6. Imbedded Markov chain of the process.

Fig. 7 .

 7 Fig. 7. Mean absorption time [in min] for N = 50, K = 3 and µ = 1 min -1 . The z-axis represents the time, whereas the initial state is represented on the horizontal plane, in barycentric coordinates with respect to each of the three absorbing states.

Fig. 8 .

 8 Fig. 8. Reaching probability for N = 50, K = 3 and µ = 1 min -1 .

1 Fig. 9 .Fig. 10 .

 1910 Fig. 9. Mean reaching time [in min] for N = 50, K = 3 and µ = 1 min -1 .The maverickness rate α grows from 0 to 0.1. Note that the scale of the z-axis varies.

2 Fig. 11 .

 211 Fig. 11. Invariant measure for N = 50, K = 2 and µ = 1 min -1 . When α exceeds 0.038, the shape of the curve reverses.

  A k , then (9) does not depend on n. By symmetry w.r.t. k, we can denote it by E[S]. Fig. 12 depicts the evolution of E[S] in function of α for several values of N . Let α min be the value of α such that E[S] is minimum. For certain values of N , E[S] is strictly decreasing over [0, 1], so that α min = 1, but for some other values of N (like N = 1000), E[S] starts increasing after α min .

Fig. 12 .

 12 Fig. 12. Mean scattering time [in min] with respect to α. Nsup/N = 80%, N inf /N = 20%, K = 3 and µ = 1 min -1 . For each curve, the vertical solid line corresponds to the critical value αc, while the dotted line corresponds to α min (whenever it is less than the right bound of the axis).

Fig. 13 .

 13 Fig. 13. Traces collected during the Fête de la Musique between the three attracting zones.

Fig. 14 .

 14 Fig. 14. Traces collected during the Fête de la Musique in the triangle Châtelet-Saint Michel-Bastille.

As N → ∞, this assumption can be interpreted as follows : since αc ∼ K/N , it means that, asymptotically, it is enough to have only one maverick per zone to saturate the system, which is quite surprising.

Disclaimer

This paper is the second version of a paper we published in Infocom, Proceedings IEEE (2010). Equations ( 5) and ( 6) (here equations ( 6) and [START_REF] Leaves | Dynamic Spectrum Allocation in Composite Reconfigurable Wireless Networks[END_REF]) have been fixed, some proofs have been simplified and there are some additional results and explanations.

APPENDIX

A. Resolution of Equation [START_REF] Camp | A Survey of Mobility Models for Ad Hoc Network Research[END_REF] 0<i<N q mi h (i,N -i) = -1 can be rewritten as :

otherwise.

Thus we have to invert the (N -1) × (N -1) matrix U = (u i,j ) with u i,i = 2, u i,i-1 = u i-1,i = -1, u i,j = 0 otherwise. One can easily check that U -1 is given by V /N , where the elements of V are :

Equation (2) follows immediately.

Let us now give an asymptotic estimate of this expression. We introduce x = n 1 /N , t = i/N and dt = 1/N . We have :

Both sums of the right handside can be interpreted as Riemann integrals on [x, 1] and [1 -x, 1]. Let us work out the first one for example :

We would obtain the same inequality replacing x by 1 -x.

Defining the remainder :

we have µr n ≤ 2. Thus, our estimate is

and the error is uniformly bounded by 2/µ.

B. Proof of equation (6)

We are now looking for a solution of equation ( 1) in K dimensions. First, let us start from equation [START_REF] Kauffmann | Measurement-Based Self Organization of Interfering 802.11 Wireless Access Networks[END_REF], in order to prove it satisfies the equation.

First remark : is one of the n k s is zero, then we can remove zone k without changing the sum in [START_REF] Kauffmann | Measurement-Based Self Organization of Interfering 802.11 Wireless Access Networks[END_REF], since the subsum indexed over 0 ≤ i < n k is empty. Furthermore, this will not change (1) either, since transitions towards empty zones are forbidden. Thus, without loss of generality we assume that all n k s are nonzero.

If K = 1, we are in an absorbing state and equation ( 6) holds. Otherwise, following equation ( 5), let us introduce

and :

whence :

Note that the second sum is well-defined, because n k = 0 and K > 1, so that n k ′ < N and N -i cannot be zero. Now, we can prove that (6) verifies equation (1) :

We still have to prove that in our case, there is no other solution of (1). In other words, we have to prove that the only solution of :

is the zero-vector. As usual, we introduce q nn = -λ n , and q nn ′ = λ n p nn ′ if n = n ′ (where the p nn ′ s are the As previously, let us begin with the case K = 2. We are in state (m, N -m) and we are looking for the hitting time h m of {n/n 1 < N inf }.

We want to obtain an analytical formula of h m (let us say for

.

Then the idea is to introduce :

.

We obtain the following value of v m -v m+1 :

Since q N,N h N + q N,N -1 h N -1 = -1, we also have :

, and :

To determine the h m s, we still miss an equation. This is the last one before reaching {n/n 1 < N inf } :

Together with h N inf +1 -h N inf = u N inf +1 , this leads successively to :

and, more generally, to :

In particular, for m = N sup + 1, we obtain the claimed formula. Now suppose we are in the general case. It is perfectly correct to consider the union of the K -1 last zones as one big zone. A sheep will stay in this "macro-zone" with probability n2+...+n k N , and a maverick will stay with probability K-1 K . This last probability is the only adaptation we have to make in the previous calculation. Introducing β ′ = (K -1)β, we obtain the following new recursion :

The rest of the calculation is exactly the same. In particular, for a fixed n, the result is the same for each n such that n 1 = n.


Frédéric Le Roux

On closed subgroups of the group of homeomorphisms of a manifold

Keywords:

Let M be a triangulable compact manifold. We prove that, among closed subgroups of Homeo 0 (M ) (the identity component of the group of homeomorphisms of M ), the subgroup consisting of volume preserving elements is maximal.

Introduction

The theory of of groups acting on the circle is very rich (see in particular the monographs [START_REF] Ghys | Groups acting on the circle[END_REF][START_REF] Navas | Grupos de difeomorfismos del círculo[END_REF]). The theory is far less developed in higher dimension, where it seems difficult to discover more than some isolated islands in a sea of chaos. In this note, we are interested in the closed subgroups of the group Homeo 0 (M ), the identity component of the group of homeomorphisms of some compact topological n-dimensional manifold M . We will show that, when n ≥ 2, for any good (nonatomic and with total support) probability measure µ, the subgroup of elements that preserve µ is maximal among closed subgroups.

Let us recall some related results in the case when M is the circle. De La Harpe conjectured that P SL(2, R) is a maximal closed subgroup ( [Bes]). Ghys proposed a list of closed groups acting transitively, asking whether, up to conjugacy, the list was complete ( [START_REF] Ghys | Groups acting on the circle[END_REF]); the list consists in the whole group, SO(2), P SL(2, R), the group Homeo k,0 (S 1 ) of elements that commutes with some rotation of order k, and the group P SL k (2, R) which is defined analogously. The first conjecture was solved by Giblin and Markovic in [START_REF] Giblin | Classification of continuously transitive circle groups[END_REF]. These authors also answered Ghys's question affirmatively, under the additional hypothesis that the group contains some non trivial arcwise connected component. Thinking of the two-sphere with these results in mind, one is naturally led to the following questions.

Question 1. Let G be a proper closed subgroup of Homeo 0 (S 2 ) acting transitively. Assume that G is not a (finite dimensional) Lie group. Is G conjugate to one of the two subgroups: (1) the centralizer of the antipodal map x → -x, (2) the subgroup of area-preserving elements?

Note that the centralizer of the antipodal map is the group of lifts of homeomorphisms of the projective plane; it is the spherical analog of the groups Homeo k,0 (S 1 ).

Question 2. Is P SL(2, C) maximal among closed subgroups of Homeo 0 (S 2 )?

On the circle the group of measure-preserving elements coincides with SO(2). It is not a maximal closed subgroup since it is included in P SL(2, R). In contrast, we propose to prove that the closed subgroup of area-preserving homeomorphisms of the two-sphere is maximal. To put this into a general context, let M be a compact topological manifold whose dimension is greater or equal to 2. We assume that M is triangulable and (for simplicity) without boundary. Let us equip M with a probability measure µ which is assumed to be good: this means that every finite set has measure zero, and every non-empty open set has positive measure. We consider the group Homeo 0 (M ) of homeomorphisms of M that are isotopic to the identity, and the subgroup Homeo 0 (M, µ) of elements that preserve the measure µ. According to the famous Oxtoby-Ulam theorem ([OU41, GP75], see also [START_REF] Fathi | Structure of the group of homeomorphisms preserving a good measure on a compact manifold[END_REF]), if µ ′ is another good probability measure on M then it is homeomorphic to µ, meaning that there exists an element h ∈ Homeo 0 (M ) such that h * µ = µ ′ . In particular the subgroup Homeo 0 (M, µ ′ ) is isomorphic to Homeo 0 (M, µ). We equip these transformation groups with the topology of uniform convergence, which turns them into topological groups. The subgroup Homeo 0 (M, µ) is easily seen to be closed in Homeo 0 (M ). Note that according to Fathi's theorem (first theorem in [START_REF] Fathi | Structure of the group of homeomorphisms preserving a good measure on a compact manifold[END_REF]), Homeo 0 (M, µ) coincides with the identity component in the group of measure preserving homeomorphisms. The aim of the present note is to prove the following.

Theorem. The group Homeo 0 (M, µ) is maximal among closed subgroups of the group Homeo 0 (M ).

In what follows we consider some element f ∈ Homeo 0 (M ) that does not preserves the measure µ, and we denote by G f the subgroup of Homeo 0 (M ) generated by {f } ∪ Homeo 0 (M, µ).

Our aim is to show that the group G f is dense in Homeo 0 (M ).

Localization

In this section we show how to find some element in G f that has small support and contracts the volume of some given ball.

Good balls A ball is any subset of M which is homeomorphic to a euclidean ball in R n , where n is the dimension of M . We will need to consider balls which are locally flat and whose boundary has measure zero. More precisely, let us denote by B r (0) the euclidean ball with radius r and center 0 in R n . A ball B will be called good if µ(∂B) = 0 and if there exists a topological embedding (continuous one-to-one map) γ : B 2 (0) → M such that γ(B 1 (0)) = B. Note that, due to countable additivity, if γ : B 1 (0) → M is any topological embedding, then for almost every r ∈ (0, 1) the ball γ(B r (0)) is good.

Oxtoby-Ulam theorem We will need the following consequence of the Oxtoby-Ulam theorem. Let B 1 , B 2 be two good balls in the interior of some manifold M ′ , with or without boundary (what we have in mind is either M ′ = M or M ′ is a euclidean ball). Let µ ′ be a good probability measure on M ′ which assigns measure zero to the boundary ∂M ′ . Denote by Homeo 0 (M ′ , µ ′ ) the identity component of the group of homeomorphisms of M ′ which are supported in the interior of M ′ and preserve µ ′ . Assume µ ′ (B 1 ) = µ ′ (B 2 ). Then there exists φ ∈ Homeo 0 (M ′ , µ ′ ) such that φ(B 1 ) = B 2 . To construct φ, we first choose a good ball B in the interior of M ′ that contains B 1 , B 2 in its interior. According to the annulus theorem ([Kir69, Qui82]), we may find a homeomorphism φ ′ supported in the ball B that sends B 1 onto B 21 . A first use of the Oxtoby-Ulam theorem provides a homeomorphism φ 1 supported in B 2 and sending the measure

(φ ′ * µ ′ ) |B 2 to the measure µ ′ |B 2 .
A second use of the same theorem gives a homeomorphism φ 2 supported in B \ B 2 and sending the measure (φ ′ * µ ′ ) |B\B 2 to the measure µ ′ |B\B 2 . Then φ is obtained as φ 2 φ 1 φ ′ . Note that, since φ is supported in the ball B, Alexander's trick ( [START_REF] Alexander | On the deformation of an n-cell[END_REF]) provides an isotopy from the identity to φ within the homeomorphisms of B that preserves the measure µ ′ , which shows that φ belongs to Homeo 0 (M ′ , µ ′ ).

Triangulations We will also need triangulations which have good properties with respect to the measure µ. We begin with any triangulation T of M . We would like the (n -1)-skeleton of T to have measure zero, but some (n -1)dimensional simplices may have positive measure. We fix this as follows. Each n-dimensional simplex s of T is homeomorphic to the standard n-dimensional simplex ; let µ s be a probability measure on s which is the homeomorphic image of the Lebesgue measure on the standard simplex. The measure

µ ′ = 1 N µ s
(where N denotes the number of n-dimensional simplices of T ) is a good probability measure on M for which the n-1-dimensional simplices have measure zero. We apply the Oxtoby-Ulam theorem to get a homeomorphism h of M sending µ ′ to µ. Then we consider the image triangulation T 0 = h * (T ), whose (n -1)skeleton has measure zero. In addition to this, all the simplices of T 0 have the same mass. Using successive barycentric subdivisions we get a sequence (T p ) p≥0 of nested triangulations with both properties: the (n -1)-skeleton have no mass and all the simplices have the same mass. Denote by m p the common mass of the simplices of T p , and by d p the supremum of the diameters of the simplices of T p (for some metric which is compatible with the topology on M ). Then the sequences (m p ) and (d p ) tends to zero.

Here is a useful consequence. Let O be any open subset of M . We define inductively O p as the set of all the n-dimensional open simplices of T p that are included in O but not in some s ∈ O p-1 . The elements of O := ∪O p are pairwise disjoint and their closures cover O. Since the (n-1)-skeleton of our triangulations have no mass, we have the equality

µ(O) = U ∈O µ(U ) (1).
We call a (closed) simplex of some T p good if it is a good ball in M . We notice that for every p > 0, all the n-dimensional simplices that are disjoint from the (n -1)-skeleton of T 0 are good2 . Thus equality (1) still holds if, in the definition of the O p 's, we replace the simplices by the simplices whose closure is good. As a consequence, if two probability measures µ, µ ′ give the same mass to all the good simplices of T p for every p, then they are equal.

In the first Lemma we look for elements of the group G f that do not preserve the measure and have small support.

Lemma 2.1. For every positive ε there exists a good ball B of measure less than ε and an element g ∈ G f which is supported in B and does not preserve the measure µ.

Proof. By hypothesis the probability measures µ and f * µ are not equal. According to the discussion preceding the Lemma, there exists some p > 0 and some simplex of the triangulation T p whose closure B 1 is a good ball, and such that µ(B 1 ) = µ(f -1 (B 1 )). To fix ideas let us assume that

µ(f -1 (B 1 )) > µ(B 1 ).
This implies the same inequality for at least one of the simplices of T p+1 that are included in B 1 ; thus, by induction, we see that we may choose p to be arbitrarily large. Note that we have µ(f -1 (M \B 1 )) < µ(M \B 1 ). Thus the same reasoning, applied to M \ B 1 , provides a (closed) simplex B 2 of some T p ′ , disjoint from B 1 , such that µ(f -1 (B 2 )) < µ(B 2 ).

Again, by induction, we may assume that p ′ = p and this is an arbitrarily large integer. In particular B 1 and B 2 are good balls with the same mass. Let B ′ be a ball whose interior contains B 1 and B 2 . Since B 1 and B 2 have the same measure, by the above mentioned version of the Oxtoby-Ulam theorem there exists φ ∈ Homeo 0 (M, µ) supported in B ′ and sending B 1 onto B 2 . Now we consider the element

g = f -1 φf of the group G f . It has support in the ball B = f -1 (B ′ ). It sends the ball f -1 (B 1 ) to the ball f -1 (B 2
), and we have

µ(f -1 (B 1 )) > µ(B 1 ) = µ(B 2 ) > µ(f -1 (B 2 ))
so that g does not preserve the measure µ, as required by the Lemma. It remains to see that in the above construction we may have chosen B to be a good ball of arbitrarily small measure. Since µ has no atom, for every ε > 0 there exists some η > 0 such that every subset of M of diameter less than η has measure less than ε. Thus by choosing p = p ′ large enough we may require that

µ(f -1 (B 1 )) + µ(f -1 (B 2 )) < ε.
Then we choose B as a ball whose interior contains f -1 (B 1 ) and f -1 (B 2 ) and which still has measure less than ε. Finally we shrink B a little bit to turn it into a good ball. This completes the proof of the Lemma.

We subdivide the euclidean unit ball B 1 (0) of R n into the half-balls

B - 1 = B 1 (0) ∩ {x ≤ 0} and B + 1 = B 1 (0) ∩ {x ≥ 0}. Let Σ be the disk B - 1 ∩ B +
1 that separates the half-balls. We consider a given ball B and some homeomorphism g supported in B. For every homeomorphism γ : B 1 (0) → B we let γ ± = γ(B ± 1 ); we say that γ is thin if γ(Σ) has measure zero. We now consider the set I(γ, g) of all the numbers of the type

µ(g(γ + )) -µ(γ + )
where γ is thin.

Lemma 2.2. If g does not preserve the measure µ then I(γ, g) contains an interval [a -, a + ] with a -< 0 < a + .

Proof. First we want to prove that there exists some γ : B 1 (0) → B which is thin and such that µ(g(γ + )) = µ(γ + ). Since g does not preserve the measure µ, we may find some good ball b in the interior of B such that µ(b) = µ(f -1 (b)). To fix ideas we assume that µ(b) < µ(f -1 (b)). Thanks to the Oxtoby-Ulam theorem we may identify B with a euclidean ball in R n , b with another euclidean ball inside B, and µ with the restriction of the Lebesgue measure on R n . All our balls are centered at the origin. Let b ′ be a ball slightly greater than b, and T be a thin tube in B \ b ′ connecting the boundary of B and that of b ′ . There exists a homeomorphism γ : B 1 (0) → B such that γ + = T ∪ b ′ . The construction may be done so that the (Lebesgue) measure of γ + is arbitrarily close to that of b, and then we have µ(γ + )) < µ(g -1 (γ + )), as wanted.

We can find a continuous family (R t ) t∈[0,1] of rotations of B 1 (0) such that R 0 is the identity and R 1 is a rotation that exchanges B - 1 and B + 1 . Setting

γ t := γ • R t , we have γ + 1 = γ - 0 = γ -.
Note that it may happen that γ t (Σ) has positive measure for some t. To remedy for this we consider γ ′ = φ • γ, where φ : B → B is a homeomorphism that fixes γ(Σ), such that the image under γ ′ of the Lebesgue measure on B 1 (0) is equivalent to the restriction of µ to the ball B, in the sense that both measures share the same measure zero sets; such a φ is provided by the Oxtoby-Ulam theorem. This ensures that γ ′

t := γ ′ • R t is thin for every t. Note that γ ′ 0 ± = γ ± 0 and γ ′ 1 ± = γ ± 1 . We have µ(g(γ ′ 1 + )) -µ(γ ′ 1 + ) = µ(g(γ ′ 0 -)) -µ(γ ′ 0 -) = (1 -µ(g(γ ′ 0 + ))) -(1 -µ(γ ′ 0 + )) = -(µ(g(γ ′ 0 + )) -µ(γ ′ 0 + )) = 0.
Thus the set I(γ, g) contains the interval

{µ(g(γ ′ t + )) -µ(γ ′ t + ), t ∈ [0, 1]}
is to use corollary 2.3 to progressively transfer some mass from the simplices s whose mass is larger than the mass of their image under g -1 , to those for which the opposite holds.

Here are some details. Given a triangulation T for which the (n -1)-skeleton has zero measure, we choose two n-dimensional simplices s, s ′ of T , and some positive ε less than µ(s); let us explain how to transfer a mass ε from s to s ′ . First assume that s and s ′ are adjacent. Then we may choose an embedding γ : B 1 (0) → s ∪ s ′ with γ(Σ) ⊂ s ∩ s ′ , γ + ⊂ s and γ -⊂ s ′ , and we apply corollary 2.3. Thus we get an element h ∈ G f , supported in s ∪ s ′ , such that µ(h(s)) = µ(s)ε, and consequently µ(h(s ′ )) = µ(s ′ ) + ε. Now consider the general case, when s and s ′ are not adjacent. Since M is connected, there exists a sequence s 0 = s, . . . , s ℓ = s ′ of simplices of T in which two successive elements are adjacent. As described before we may transfer mass ε from s 0 to s 1 , then from s 1 to s 2 , and so on. Thus by successive adjacent transfers of mass we get some element in h ∈ G f that transfers mass ε from s to s ′ . Note that the masses of all the other elements do not change, that is, µ(h(σ)) = µ(σ) for every simplex σ of T different from s and s ′ . Now we go back to our triangulation T 0 , and we construct g ′ the following way. If each simplex s has the same measure as its inverse image g -1 (s) then there is nothing to do. In the opposite case there exists some simplex s of T 0 such that µ(s) > µ(g -1 (s)). We also select some other simplex s ′ such that µ(s ′ ) = µ(g -1 (s ′ )), and we use the previously described construction of a homeomorphism g 1 ∈ G f that transfers the mass µ(s)-µ(g -1 (s)) from the simplex s to the simplex s ′ . After doing so the number of simplices g 1 (s) ∈ g 1 * T 0 whose mass differs from the mass of g -1 (s) has decreased by at least one compared to T 0 . We proceed recursively until we get an element g ′ ∈ G f such that µ(g ′ (s)) = µ(g -1 (s)) for every simplex s in T 0 , as wanted for this first step.

For the second and last step we consider the triangulations (g -1 ) * (T 0 ) and g ′ * (T 0 ). The homeomorphism g ′ g sends the first one to the second one, and each simplex g -1 (s) ∈ (g -1 ) * (T 0 ) has the same measure as its image g ′ (s) ∈ g ′ * (T 0 ). We apply Oxtoby-Ulam theorem independently on each g ′ (s) to get a homeomorphism Φ s : g ′ (s) → g ′ (s), which is the identity on ∂g ′ (s), and which sends the measure (g ′ g) * (µ |g -1 (s) ) to the measure µ |g ′ (s) . The homeomorphism Φ := s Φ s g ′ g preserves the measure µ. Furthermore by Alexander's trick each Φ s is isotopic to the identity, thus Φ is isotopic to the identity, and belongs to the group Homeo 0 (M, µ). Now the homeomorphism g ′′ = g ′-1 Φ belongs to the group G f and for each simplex s of the triangulation T 0 we have g ′′-1 (s) = g -1 (s). We may have chosen the triangulation T 0 so that each simplex has diameter less than some given ε. Every point x in M belongs to some n-dimensional closed simplex g -1 (s) of the triangulation (g -1 ) * T 0 , and since both g(x) and g ′′ (x) belong to s they are a distance less than ε apart. In other words the uniform distance from g to g ′′ is less than ε. This proves that g belongs to the closure of G f , and completes the proof of the theorem.

One may probably avoid the use of the annulus theorem here, since the ball B may be constructed explicitly by gluing the two good balls B1 and B2 to a piecewise linear tube connecting them.

Note that there may be simplices in T0 that fail to be good balls if T0 is a triangulation but not a PL-triangulation.

which contains both a positive and a negative number, as required by the lemma.

Corollary 2.3. Let γ 0 : B 1 (0) → M be a topological embedding in M with µ(γ 0 (Σ)) = 0, let B 0 = γ 0 (B 1 (0)), and let ε > 0 be less than the measure of γ + 0 . Then there exists some element g ∈ G f , supported in B 0 , such that

In the situation of the corollary we will say that g transfers a mass ε from γ + 0 to γ - 0 .

Proof. Lemma 2.1 provides some element g ′ ∈ G f that does not preserve the measure µ, and which is supported on a good ball B whose measure is less than the minimum of µ(γ + 0 )ε and µ(γ - 0 ). Then Lemma 2.2 provides some homeomorphism γ : B 1 (0) → B which is thin and such that g ′ transfers some mass a from γ + to γ -:

Since such a number a may be chosen freely in an open interval containing 0, we may assume that a = ε N for some positive integer N . Choose some homeomorphism Φ 1 ∈ Homeo 0 (M, µ) that sends B inside B 0 , γ + inside γ + 0 and γ -inside γ - 0 . Such a Φ 1 is provided by Oxtoby-Ulam theorem, thanks to the fact that we have chosen the measure of B to be small enough and that µ(γ(Σ)) = µ(γ 0 (Σ)) = 0. Now the conjugate g 1 = Φ 1 g ′ Φ -1 1 transfers a mass a from γ + 0 to γ - 0 :

We repeat the process with γ 1 = g 1 • γ 0 instead of γ 0 , getting an element g 2 ∈ G f that tranfers a mass a from γ + 1 to γ - 1 :

We repeat the process N times, and get the final homeomorphism g as a composition of the N homeomorphisms g N , . . . , g 1 .

Proof of the theorem

We consider as before some element f ∈ Homeo 0 (M ) \ Homeo 0 (M, µ). Let g be some other element in Homeo 0 (M ). In order to prove the theorem we want to approximate g with some element in the group G f generated by f and Homeo 0 (M, µ). We fix a triangulation T 0 for which the (n -1)-skeleton has zero measure. The first step of the proof consists in finding an element g ′ ∈ G f satisfying the following property: for every simplex s of T 0 , the measure of g ′ (s) coincides with the measure of g -1 (s). To achieve this, the (very natural) idea