M Al

Heib Ineris

M Degas

Post-Mining Risk Management in France and multi-hazard approaches for coal mines

Keywords: post-mining, multi-hazard, interaction, natural hazard, matrix, adjustment

After centuries of intensive exploitation of its mineral resources, French mining sites have gradually closed. However, after mine-closure and during the "post-mining" phase, numerous residual hazards can occursometimes as soon as mining works stop, or later, such as: ground movement phenomena (subsidence, collapses), rising gas, irreversible disruptions in underground water circulation induced by mining can potentially cause disturbances, both in terms of water circulation (flooding in low areas, disruption of waterway flows) and water quality (pollution).

To manage hazards and risks associated with these undesirable events, the French State has several technical and regulatory tools at its disposal. These tools make it possible to compile the available knowledge on residual mining risks related to former mining sites for a given territory, to delimit the affected areas and to define the conditions of construction, occupation, and use of land as well as measures relating to the organization, use or exploitation of existing assets in a context of a sustainable land planning management. This paper presents the methodology to assess the post-mining hazards assessment. The paper also focuses on the development of new methodology for multi-hazards analysis regarding post-mining, natural and technologic hazards. The methodology analyses the hazards interactions and consequences on the environment. The multi-hazards assessment methodology consists of three steps: the identification of the singles hazards, the identification of the hazard interactions and finally the identification of the level and the consequences of the interactions. The matrix tool and interaction organigrams are used to identify the potential interactions. Three levels of interaction are considered: simple interaction, double interaction and dominos or cascading interactions. The natural and mining flooding hazard seems as the main hazard that can trigger several mining hazards: such as ground movement (subsidence, landslide, gas production, etc.).

Introduction and objectives

The former mining sites are generally associated with several hazards and risks [START_REF] Parry | Abandoned mine workings manual, C758D[END_REF]. Thus, risk assessment and their management are a central objective of the mining industry, mining authorities and decision-makers. Thus, the post-mining risk assessment presents a challenge to be in line with the sustainable development of mining regions. Mine closure can result in hazards, such as ground movements, induced or natural seismicity, hydraulic perturbations, flooding events, dangerous or toxic gas emissions or releases of potentially dangerous chemical substances into the environment [START_REF] Bell | Mining and its impact on the environment[END_REF][START_REF] Bell | Mining and its impact on the environment[END_REF][START_REF] Ch | Mine closure and post-mining management[END_REF]. The assessment of the different hazards must consider the interaction between mining hazards, i.e. the influence of one hazard on others. Multi-risk assessment research started relatively recently, in the 1990s, around natural hazards such floods, earthquakes, landslides, volcanic eruptions, snow avalanches and their potential interactions over a territory [START_REF] Gallina | A review of multi-risk methodologies for natural hazards: Consequences and challenges for a climate change impact assessment[END_REF]Garcia et al., 2016, Komendantova et al, 2015;[START_REF] Garcia-Aristizabal | Multi-risk Assessment as a Tool for Decision-making[END_REF]. Multi-risk assessment is then developed as a decision-support tool for climate change and urban vulnerability [START_REF] Garcia-Aristizabal | Multi-risk Assessment as a Tool for Decision-making[END_REF]. Therefore, the interest in multi-risk assessment increased in the last decades, especially when it comes to applications and initiatives to assess risks from different natural and anthropogenic hazard events [START_REF] Gallina | A review of multi-risk methodologies for natural hazards: Consequences and challenges for a climate change impact assessment[END_REF]. A new risk assessment discipline was developed in recent years concerning the interaction between natural and technological hazards, so-called Natech, requiring a comprehensive understanding of the interdependencies of human, natural and technological hazards [START_REF] Krausmann | NaTech Risk Assessment and Management: Reducing the Risk of Natural-Hazard Impact on Hazardous Installations (1st Edition)[END_REF]).

In the mining context, the risk and hazard assessment studies have focused on a single hazard than multihazards [START_REF] El Shayeb | Back analysis for predicting type and size of subsidence hazard over abandoned Lorraine iron mines[END_REF][START_REF] Heib | Residual subsidence analysis after the end of coal mine work[END_REF][START_REF] Abdul-Wahed | Mining-induced seismicity: seismic measurement using multiplet approach and numerical modelling[END_REF][START_REF] Morgan | An analysis of water risk in the mining sector[END_REF]Dobson, 2020 Zhao and[START_REF] Zhao | Land subsidence risk assessment and protection in mined-out regions[END_REF]. However, closed mining areas are generally not affected by a single mining or natural hazard, but two or more can act at the same time or consecutively [START_REF] Merad | Use of Multi-Criteria Decision-Aids for Risk Zoning and Management of Large Area Subjected to Mining-Induced Hazards[END_REF][START_REF] Lenhardt | The Impact of Earthquakes on Mining Operations. BHM Berg-und Hüttenmännische Monatshefte[END_REF]Kim et al., Azhari et al., 2017[START_REF] Mavrommatis | Towards a comprehensive framework for climate change multi-risk assessment in the mining industry[END_REF]. In this context, assessing a single hazard can be unmanageable when multiple hazards must be considered. Furthermore, in a post-mining context, a multi-hazard approach is not apparent: the available data for the different single hazards may refer to different spatial scales; comparisons, rankings and aggregations can be difficult; different specialised entities and experts need to collaborate.

The objective of this paper is to present the existing methodology used in France to assess a single mining hazard, then the approach used to identify the interaction between several hazards.

Definitions and methodology

The term multi-hazard refers to the existence and the occurrence of several hazards in the same territory. The multi-risk assessment objective is to better consider the consequences of the interaction between hazards and risks. Hence, it can lead to improve the management of mining operations by authorities, stockholders, and decision makers of mining regions. [START_REF] Delmonaco | Report on a new methodology for multi-risk assessment and the harmonisation of different natural risk maps[END_REF] define multi-hazard analysis as the 'implementation of methodologies and approaches aimed at assessing and mapping the potential occurrence of different types of natural hazards in a given area'. The European Commission (2010) considers multi-hazard analysis as the probability of occurrence (the probability of occurrence can be used to quantify a specific hazard) of different hazards, either occurring at the same time or shortly following each other, because they are dependent on each other or because they are caused by the same triggering event, such as rainfall, earthquake hazard or merely threatening the same elements at risk without chronological coincidence. [START_REF] Liu | A three-level framework for multi-risk assessment, Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards[END_REF] proposed a three-level framework for multi-risk assessment that considers the possible risk interactions. The first level concerns the existing or not interaction between the identified hazards. The second level is a semi-quantitative analysis approach to determine whether a more detailed, quantitative assessment is needed. The third level of the methodology is a detailed quantitative multi-risk analysis.

In this context, a multi-hazard assessment should be adopted as risk assessment method. The available data for the different single hazards should be analysed; compared, ranked, and aggregated by experts to identify the potential interaction and the level of the interactions. [START_REF] Mavrommatis | Towards a comprehensive framework for climate change multi-risk assessment in the mining industry[END_REF] present a comprehensive framework for multi-risk analysis of climate change and provide operational recommendations for managing the interaction between the mining industry and natural hazards related to climate change.

Advantages and limitations

The reason of the development of multi-hazard approach is to reduce the cost of hazards. Feedback from various European countries has shown that separate hazard management decreases the effectiveness of prevention, because it does not consider the effects of interactions between phenomena and the effects of hazard-risk interactions. It, therefore, appears more and more necessary to consider risks in a 'globa' approach, whereas mine managers and local authorities often manage only single hazards. In principle, single-hazard approaches assess the hazards separately, which implies that the solutions provided for their management do not consider the other phenomena and are sometimes incompatible with them. When the analysis does not take into account the interdependencies between the hazards, the assessment presents tools of little relevance to managing complex risks likely to lead to regulatory contradictions. For a site exposed to several hazards, multi-hazard analyses, unlike single-hazard analyses, involve considering each hazard as an element potentially interacting with other hazards. The comprehensive and integrative approach of multi-risk analysis, which considers several hazards and the associated vulnerabilities, best represents situations where several hazards coexist and often interact on the same territory/site. The main advantages of multi-hazard of closed mines are:

• Improvement of the quality of the risk assessment analysis.

• Identification of the scenarios related to their interactions.

• Better considering the vulnerabilities of a territory exposed to several hazards.

• Improvement of the preservation of the general interests identified around closed mines.

• Improvement in the resilience capacity and sustainability of the territories.

However, multi-hazard risk assessment at local and regional scales remains a significant challenge due to the lack of data, causal factors, and interactions between different types of hazards [START_REF] Touili | La gestion des risques multiples en zones urbaines : un modèle intégré d'analyses multirisques pour une résilience générale[END_REF]. Multi-risk assessment tools can help decision-makers and provide them with information on mitigation measures [START_REF] Komendantova | Multi-hazard and multi-risk decision-support tools as part of participatory risk governance: Feedback from civil protection stakeholders[END_REF].

Hazard interaction identification and methodology

The assessment of the physical interactions is based on the qualification of the hazard intensity (high level, average level, low level) and the predisposition of each single hazard. We distinguish between three types of interaction: coupling, depending, and cascading. For the coupling interaction, the first hazard amplifies the second hazard. For the depending hazards, only the second hazards can occur if the first hazard occurred. Finally, the cascading interactions occurred when the first hazard triggers several hazards (cascading effect) when several hazards have the same triggering factors and initiating events. In addition, the hazard modifies the conditions of one or more hazards. When one hazard occurs, the conditions for a second hazard may be met, the area becomes vulnerable, or the probability of occurrence becomes higher. The second hazard is entirely or partially related to the first one. A full multi-hazard assessment should consider all the possible hazard sources and identify all possible interaction scenarios, including cascading effects [START_REF] Garcia-Aristizabal | Multi-risk Assessment as a Tool for Decision-making[END_REF]. The interaction between hazards should consider both temporal and spatial scales. The spatial scale refers to the area where the hazard has an impact, and the temporal scale refers to the time scale during which the single hazard acts on the natural environment [START_REF] Gill | Reviewing and visualising the interactions of natural hazards[END_REF].

To assess and presents the hazard interactions, different tools are generally used. Among them: the hazard matrix, fault trees [START_REF] Eshrati | Multi hazards risk assessment, a new methodology[END_REF]; multi-criteria analysis [START_REF] Sigtryggsdottir | Methodology for geohazard assessment for hydropower projects[END_REF][START_REF] Merad | Use of Multi-Criteria Decision-Aids for Risk Zoning and Management of Large Area Subjected to Mining-Induced Hazards[END_REF][START_REF] Mladineo | Project management in mine actions using a Multi-Criteria-Analysis-based decision support system[END_REF]; the negotiated choice; the implementation of a multi-scale GIS (Global Information Geographic) and statistical modelling of vulnerability including temporal variability [START_REF] Mancini | GIS-Based Assessment of Risk due to Salt Mining Activities at Tuzla (Bosnia and Herzegovina)[END_REF]Sklodimou et al., 2019) are used to assess the multi-hazards. Additionally, the experts opinion and feedback are generally used to assess the interaction between hazards (Figure 1). The interaction matrix presents the interrelation between n hazards (Hi to Hn), for instance, the source hazard (Hi) can trigger several hazards (H, i to n). Figure 1-a presents the interaction modes between two hazards. Figure 1-b andc, present the construction and mapping of the interaction matrix. The index Iij presents by colure the level of the interaction. The Figure 1-c presents the matrix for five hazards with the green colour means the interaction level is very low and limited, the orange colour means that interaction level is moderate and red colour is very important and has consequences in term of risk assessment.

Hi: the hazard I, it can be considered as the resource hazard, Hj: is the trigger hazard. I1n, the interaction level (green, orange and red) of the hazards HI and Hj.

Multi-hazard assessment in a mining area

Hazards categories in a mining area

In a mining area, the main hazards likely to occur are grouped into 3 large families for which the assessment methods are different: mining hazards (M), natural hazards (N) and technological hazards (T). Mining hazards are generally gathered into 6 groups: ground movements; combustion and fire in mine deposits and dumps; hydrological and hydrogeological disturbances of mining origin; gas emissions related to mining; endogenous radioactivity of the environment; and environmental pollution, which may impact on water, soil and air.

Natural hazards are related to the environment such as floods, wildfires, etc. [START_REF] Djizanne | Development of post-mining multi-hazard assessment methodology[END_REF].

Technological hazards are related to industrial activities such as industrial pollution, toxic wastes, dam failures, etc. Table 1 summarize of the different hazards can be identified in a closed mining site [START_REF] Djizanne | Development of post-mining multi-hazard assessment methodology[END_REF].

Mining hazard classification

To assess a single mining hazard two stages are carried out:

• an "information" stage consisting of a description of the mining sites studied (brief history, geographic and geological environment, form and layout of exploitation, inventory of past disturbances) and the collection and evaluation of archive and land data needed to locate and evaluate the hazard. At the end of this stage, one or more informative maps are produced.

• a hazard evaluation stage which defines, for each phenomenon identified as relevant for the sites studied and for each mining configuration, the intensity and predisposition criteria described above and the severity level of the hazard. At the end of this stage, one or more hazard maps are produced based on the number of relevant phenomena and the scope of the territory studied. The hazard study report brings these two stages together. In the mine conditions, three steps are required to qualify the mining hazard: the qualification of the intensity of the hazard and the predisposition. The hazard level is multiplication of hazard intensity by the predisposition factors.

Qualifying intensity classes

The approach developed in France [START_REF] Salmon | Post-Mining Risk Management in France[END_REF] to assess the intensity of a phenomenon consists in identifying the most representative physical parameters in order to characterize the consequences of potentially dangerous events. Thus, one can choose to focus on criteria related to the size of collapse craters, the amplitude of horizontal surface land deformations or the nature, content, and flow of gaseous emissions, etc. Characterizing potential consequences involves referring to the concept of the "severity" of potential events. Severity means the extent of foreseeable consequences to targets that may be present on the surface. This can apply to people (victims), ecological damages and property damages. The number of intensity classes used for analysis may vary based on the context of the study and especially the accuracy and exhaustiveness of the input data. Hazard studies carried out of post-mining risk use the following classes: very low (rarely used, reserved for phenomena with very low occurrences), low, moderate, high and very high (also rarely used, reserved for devastating events of exceptional intensity).

Qualifying predisposition classes

Qualification of a predisposition consists of an analysis of the possibility that a phenomenon will appear or manifest on the sub-surface. This analysis is based first on experiential feedback, i.e. on past occurrences of disturbances or nuisances on the site being studied or on a similar site. But a mining site that may not have been the location of known disturbances (some may have been forgotten) may nonetheless feature favourable conditions for a disturbance to occur. Thus, the second approach is to detect these mine configurations by examining the type and configuration of the mining works and their topographical, geological and hydrogeological environment. In addition, because most of the mines in France are very old, it is very rare to have access to all the documents and plans on works, structures and previous mine disorders. Furthermore, some of these documents and plans contain inaccuracies or are based on references that no longer exist. Because of the uncertainties generated by this incomplete and fragmented information, a predisposition analysis may include a criterion for the presumed presence of mining works and/or structures that may point to the presence of a hazard. Thus, this is a complex approach that requires hazard. The predisposition classes are: very unlikely (rarely used), unlikely, likely, very likely.

Qualifying hazard classes

Both implicit and explicit approaches are used to combine qualitative values amongst themselves or to cross reference qualitative and quantitative criteria. These may include techniques that use scoring systems, rankings, multi-criteria classification, etc. If the two-way table system is selected, use a matrix like the one illustrated as an example in the table below (Table 2), keeping in mind that each site may require adjustments to fit its specific context. Hazard level is evaluated on a case-by-case basis for each site. The following terminology should be used to qualify the three hazard classes: low, medium, and high.

Mining hazards interactions

The methodology of the multi-hazard assessment is divided into three main steps:

The first step describes the three significant hazards families: mining, natural, and technological. For the considered site, the singles hazards should be identified based on the characteristic of the hazards and the related external factors. For instance: the sinkhole hazard depends on the depth, the dimensions of the underground cavity, the strength of the upper layer. The external factors, for sinkhole hazard, are the flooding, the traffic, the aging, etc.

The second step analysis the potential interaction based on the common factors of the hazards and conditions of the occurrences of the hazards. Possible interactions between hazards are based on the following: their nature (triggering or aggravating), their category (physical or regulatory), and their typology (dependent or independent). The visualisation of the potential interaction is obtained using matrix interaction tool and/or the diagram tool (cf. Figure 1 and Figure 4).

The third step is focused on the identification of the level of interactions between hazards. The level of the interaction is based on the intensity of the single hazards and the level of the interaction.

More concretely, for a mining site, the following questions should be answered by the experts in charge of study to identify the potential interactions between the different hazards:

Interaction conditions: are there specific conditions to be fulfilled? What are these conditions? How to evaluate their likelihood? Or is the interaction systematic?

Intensity: to what extent should a specific source phenomenon modify the target phenomenon intensity? What are the parameters that explain target phenomenon intensity?

Probability of occurrence: which parameters should modify the target probability of occurrence of the phenomenon?

Temporality: will the source and target phenomena coincide, or is there a buffer time between their occurrences? What are the parameters influencing the buffer time?

It Is necessary to carry out a collect of the information related to the mining site. When a lake of data is meted, experts generally can use the information from equivalent sites having the comparable conditions.

Based on the feedback analysis, the following examples of interaction are identified for a mining hazard as a trigger to another mining/natural/technology hazard(s), (Table 3).

The interaction between mining, natural and technology hazards, depends on the scales: spatial scale and temporal scale. Figure 2 2 represents the potential interaction of several mining-natural hazards using a temporal and spatial scales. The spatial scale covers very limited surface (very local) to large surface (regional land). The temporal scale covers very short event, hours, to very long period (years). Certain mining hazards are very local and very short (e.g.: a sinkhole hazard), the interaction with another hazard may be limited even the interaction theoretically is possible. The interaction should fulfil the following conditions: the occurrence of the H1 corresponds, in time and in space, to the occurrence of the hazard H2. For instance, the flooding of a mining site, large surface, can interact with the sinkhole hazard, if and only if the collapse of the cavity is imminent or shortly can be happened. In this case the level of the interaction between the two hazards can be considered as high.

In the other hand, certain hazard can concern a large surface (hectares) and can last a long time (years): selffire or self-combustion of coal dump. Under specific condition, long drought period, the coal can start the self-heating. Thus, the self-heating hazard can trigger a pollution of water and air for a long distance, etc. In this example, it is very important to assess, not only the potential of the interaction, but also the scales of the interaction (spatial and temporal).

Figure 2. Mining hazard interaction over temporal and spatial scales

An example of interaction matrix of mine hazards (see Table 1) is built for mining and natural hazards, the judgement of the level of the interaction is based on the predisposition factors and the intensity level (Table 4). Three intensity classes are considered (limited, medium and high) and three predisposition classes (likely sensitive, sensitive and very sensitive).

. Tentative view of interaction matrix of different hazards related to mine closure

To illustrate the complexity and multiple-interaction possibilities, Figure 3 presents an example of follow organigram where the local collapse hazard (sinkhole-SIN). The figure presents the theoretical potential interaction, each site must be analysed as a unique configuration. The local collapse (sinkhole, SIN) hazard can interact with 9 mining hazards and 6 natural hazards and 2 technological hazards. Furthermore, the sinkhole hazard depends on external factors such as ageing of rockmass (Age), the traffic (Tra) and the overload of backfill material or others (Sur). We can identify 22 potential interaction covering double, simple and cascading interaction. Different scenario should be assessed based on the local site conditions. The seismicity (NSI), flooding (FLO), overload, dam collapse, and others can increase and aggravate the sinkhole hazard level directly or indirectly due to the ageing phenomenon, which decreases the strength of the geomaterial. Thus, assessing the potential interactions requires significant effort to collect the different information. This illustration of the interaction can be built for all the mining hazards listed in Table 1. Based on the classification presented in 2.2, one can noticed two types of interactions: depending and cascading types. For instance: the subsidence depends on the occurrence of the flooding hazard. However, the sinkhole hazard interaction with flooding can be a cascading type because the flooding can trigger the sinkhole hazards. Such analysis can be done for the all hazard interactions. After the identification of the hazard interaction, an adjustment of the level of hazards is mandatory. Line et al. propose to adjust the level of the initial natural hazard based on the level of the interaction. Based on this statement, we adopted the same method for the mining-mining hazard interaction and mining-natural hazard interaction. Table 5 presents the initial hazard level and the adjusted hazard. Three level of interaction are considered (low, medium, and high).

Application of the Methodology to a case study

The methodology presented previously needs to be applied on a real case study. The chosen case study is a closed coal mine in France). Near the surface, above the coal mine, there is an abandoned underground limestone mine. The risk assessment studies were carried out after the shutdown of the mining activities [START_REF] Djizanne | Development of post-mining multi-hazard assessment methodology[END_REF]. The 11 mining and natural hazards are presented with their corresponding level in the Table 6. The interaction matrix was built (Table 7) based on the assessment of the factor of each hazard (Table 4, Al [START_REF] Djizanne | Development of post-mining multi-hazard assessment methodology[END_REF]. This first analysis demonstrated that the flooding hazard (FLO), due to the natural flooding (e.g., heavy rain) can trigger several mining hazards: subsidence (SUB), settlement (SET), landslide (LSG), sinkhole (SIN) and induced seismicity (INS). In addition, the flooding and the water fluctuation can increase the ground movement intensity or level, decrease the strength parameters, and mobilise the faults and discontinuity displacement. The flooding hazard can create a cascading effect. For instance, natural seismicity can trigger flooding and landslide of dumps. One can notice that the flooding (FLO) of the closed coal mine has a high interaction level. In the other hand the wildfire (FFI) has very limited interaction with the flooding (FLO). However, this conclusion should be analysed carefully more precisely based on the single hazard maps to identify the location of the interaction. In this paper, additional analysis carried out to precise the number and the level of interaction for the 6 mining hazards. We used the interaction matrix, and the flow organigram tools for identify the level of the interaction. We focus the analysis on the mining hazards. For each mining hazard, we calculate the total number of interactions (e.g. 7 for the subsidence). In all, we have identified 55 potential interactions for the six mining hazards (Table 8), a number relatively important of interactions. The Table 8 classes the hazards based on the level of interaction, the first is the sinkhole and the last one (class 5) is the subsidence hazard.

The sinkhole mining hazard (SIN) and the induced seismicity (INS) have the maximum number of potential interactions, respectively 11 and 10. However, the high level of interaction (9) concerns the sinkhole hazard (SIN). That means, the sinkhole occurrence can influence and be influenced many natural and mining hazards.

The potential interactions of the mining and natural hazards are possible if they coincide in the same place Based on the total number of interactions for each mining hazard, we classified them, and we suggested the adjustment of the initial level of the 6 mining hazards (Table 6 andTable 9). To adjust the hazard level, we used the high interaction. The following rules are adopted: if the high interaction number is less than 3, no modification of the initial hazard level, between 3 and 6, the initial hazard adjusted by one level (low becomes moderate, moderate become severe). For number of interactions bigger than 6, the initial hazard adjusted by two level (low becomes severe, moderate become severe). For the case study, we noticed that four mining hazards should be adjusted. The sinkhole hazard passes from low and moderate level to severe because of the number of high interactions >6. That means for the zones concerned by the sinkhole, we should be verified the existing of the other mining and natural hazards. For the subsidence, landslide and induced seismicity hazards, the initial level passes from low to medium, and from medium to severe.

Conclusion

The work presented in this paper concerns the development and application of a multi-hazard methodology to assess the risk of post-mining areas. The paper begins with a brief presentation of the methodology used in France to assess a single mining hazard. The methodology describing the steps involved in analysing the interaction between natural and mining hazards follows. The matrix of hazard interaction is presented for the interaction between mining-mine hazards, mine-natural hazards, and natural-mine hazards. The interaction organigram tool is used for each primary hazard (mining, natural hazard and technology). This tool completes the matrix of hazard interaction. The methodology focuses also on the scale (spatial, temporal) of the interaction They should be considered for a multi-hazard assessment.

The multi-hazard assessment presents a real advantage for mining regions where this interaction can be significant. However, the stakeholders need to create a group of experts capable to assess the interaction of hazards. The potential consequences of assessing each single hazard separately, should be study for each case study, it can increase the cost of the mitigation of hazards, and in specific cases creating the catastrophic scenario with severe social and economic consequences.

A first real-life case study of a closed coal mine in France was applied. For this case study, 6 mining hazards and 5 natural hazards were identified and assessed. Additional case studies should be used to improve the multi-hazard methodology.

The multi-hazard methodology developed here should also be improved in the frame of the research projects carried out in the mining and post-mining sector. Also, the perspective is to consider the multi-hazard and the multi-risk assessment as the main tool for case studies throughout Europe and elsewhere. Another indicator of success is therefore their adoption as part of wider projects, networks, and dialogues. This methodology must continue to evolve to much higher levels to effectively manage hazards having multiple impacts on past mining land use.

Figure 1 .

 1 Figure 1. Different tools for assessing the potential hazards of interaction. a presents the three modes of interactions, b and c present the construction of the matrix of interaction between hazards

Figure 3 .

 3 Figure 3. Examples of mining, natural and technology hazards interaction

Table 1 . Summary of the mining, natural and technological hazards used in this multi-hazard analysis (Djizanne et al. 2023)

 1

	Mining hazards (18) SUB	Subsidence Sinkhole	SUB SIN		Gas explosions Slick fire (liquid)	EXP FEN
	Crevasse	CRE	Dissolution	DIS		Flare fire (gas or liquid)	FET
	Sinkhole	SIN	Clay settlement	SET		Solid fire (solids)	FES
	Massive mine collapse	MMC	Deep landslide	DLS		Boil over	BLO
	Settlement	SET	Shallow landslide	SLS	Flammable liquefied gases	BLV
	Landslide	LS	Erosion Mudflow	ERO MUF		Liquid product release Gaseous product release	RPL RPG
	Erosion	ERO	Rocky landslide	RLS	gas	Release of a liquefied	RGL
			Rock or block fall	RFA		Fire	IPT
						Release of radioactive
	Heating	COM	Avalanche	AVA	radiation substances or nuclear	RSR
			Earthquake	NSI	Discharge of water bodies	RME
	Mine gas	GAZ	Forest fire (wildfire)	FFI	Land movement due to human activities	MVT
	Modification of the groundwater discharge regime	MWR	Settlement, consolidation	SET	Tank burst (Pneumatic energy release)	EBC
	Modification of the regime of a river	MOR	Lowland flooding		VCE (Combustion of gases, vapours)	VCE
	Flooding of topographic low points	TFL	Flooding by runoff and mudslides	FLO	Explosive vaporisation of boiling liquid)	BLV
	Flash flooding -submergence	FFS	Flooding by rising groundwater			An explosion of solids	ENA
	Induced seismicity	INS

Table 2 . Mining hazard qualification based on the qualification of the predisposition and the intensity qualification Intensity Predisposition Unlikely Likely Highly likely Low Moderate High

 2

Table 3 . Examples of interaction where the primary hazard corresponds to the hazards can trigger with secondary hazards (mining, natural and technology)

 3

	Hazard	Primary hazard			Secondary hazards	Ref.
		A collapse of the underground	Subsidence			Slope	instability,
		mine (e.g., galleries)					flooding
	Mining	flammable) Rising mine	gas	(Toxic,	environmental consequences Health	and	Fire
	hazard	inflow of fresh water A massive and uncontrolled	cavern by dissolution Collapse of a salt	seismicity Subsidence,	induced	43
		Mine flooding			Collapse			Subsidence,	slope
								stability, gas,
		Drought			hydrological		and	Gas	flow,	ground
					hydrogeological		movement,
					disturbances
		Runoff,			surface	flooding,	Subsidence, sinkhole,	Chang et
	Natural				tables increasing	water	instability, seismicity Mine collapse, slope induced	al. (2022)
	hazards	Wildfire			Pollution			Self-combustion, landslide
		Earthquake			Landslide,		fire,	Self-combustion,	(Lenhardt,
					flooding			landslide,	induced	2009,
								seismicity,	Azhari, A.;
								Ozbay,
								2017)
	Technology hazards	Dam collapse, sewerage or drinking water networks Explosion of an industrial site	Flooding, increasing water tables Wildfire, pollution	Subsidence, sinkhole, Mine collapse, slope instability, induced seismicity Landslide, self-combustion,

Table 4

 4

Table 5 . Example of adjusted hazard level considering the multi-hazard analysis: hazard interaction

 5

	Initial hazard level	Interaction level	Adjusted hazard level
	Low / Medium / High	Low / No interaction	Low / Medium / High
	Low		Medium
	Medium	Medium	High
	High		High
	Low		Medium
	Medium	High	High
	High		Very High

Table 6 . Coal mine -intensity level (low=green, moderate=orange, severe and very severe=red) of the mining hazards (6) and natural hazards (5)

 6

		Hazard	Low	Medium	Severe
		Sinkhole (SIN)
		Subsidence (SUB)
	Mine hazards	Landslide (LSG)
	(6)	Settlement (SET)
		Combustion (COM)
		Induced seismicity (INS)
		Sinkhole (SIN)
	Natural hazards (5)	Clay shrinkage -swelling (SET) Natural seismicity (NSI) Flooding (FLO)
		Wildfire (FFI)

Table 7 . Multi-hazards interaction matrix and assessment of the level of the interaction: red high, yellow: medium, green: low Source hazards Trigger hazards -mining hazards

 7

				Code	SIN	SUB	LSG	SET	COM	INS
			Sinkhole	(SIN)
		Mining	Subsidence Landslide Settlement	SUB LSG SET
			Combustion	COM
			Induced-seis	INS
			Sinkhole	SIN
	hazards	Natural	Settlement (Clay) Natural-seis Flooding	SET NSI FLO
			Wildfire	FFI

Table 8 . Potential hazard interaction between flooding (natural and mining) and mining and natural hazards. Red arrow: high interaction, low green interaction .

 8

Table 9 . Initial and adjusted mining hazard level based on the total number of the interaction

 9

Acknowledgement: This work has received funding from the European Union's Research Fund for Coal and Steel (RFCS) under the European project 'POMHAZ, https://www.pomhaz-rfcs.eu/' grant agreement No 101057326 and supported by the Ineris programs.

Code Natural hazards (17) Code Technological hazards (17) Code Subsidence


Carlo Rovelli
email: rovelli@cpt.univ-mrs.fr

Simone Speziale
email: simone.speziale@roma1.infn.it

On the perturbative expansion of a quantum field theory around a topological sector

The idea of treating general relativistic theories in a perturbative expansion around a topological theory has been recently put forward in the quantum gravity literature. Here we investigate the viability of this idea, by applying it to conventional Yang-Mills theory on flat spacetime. We find that the expansion around the topological theory coincides with the usual expansion around the abelian theory, though the equivalence is non-trivial. In this context, the technique appears therefore to be viable, but not to bring particularly new insights. Some implications for gravity are discussed.

Introduction

Conventional quantum field theory (QFT) treats interacting theories in a perturbative expansion around the quadratic term of the action. This procedure can be applied to general relativity (GR) by expanding the metric field around a fixed background, yielding the background-dependent perturbative quantum gravity of Feynman, DeWitt, Veltman and many others. The theory is non-renormalizable [START_REF] Goroff | The Ultraviolet Behavior Of Einstein Gravity[END_REF], and the approach makes sense only for computing low energy gravitons scattering. The question is open whether the nonrenormalizability is an intristic feature of the theory -and GR must be modified to be a consistent theory-, or it has to do with the perturbative expansion chosen. In this case, GR could be non-perturbatively renormalizable, or even finite [START_REF] Rovelli | Quantum Gravity[END_REF]. In this case however, a perturbative expansion, different from the standard one, could still be well defined.

An implementation of this idea has been recently proposed in [START_REF] Freidel | Quantum gravity in terms of topological observables[END_REF], based on the fact that GR can be written as a modified BF theory. BF is a topological field theory that can be quantized exactly and is finite, as first showed by Witten in 3d [START_REF] Witten | 2+1)-Dimensional Gravity As An Exactly Soluble System[END_REF]. The relevance of topological theories also for 4d quantum gravity is suspected since long [START_REF] Rovelli | The Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum gravity model in the loop representation basis[END_REF], and has been exploited in a number of ways. The idea of [START_REF] Freidel | Quantum gravity in terms of topological observables[END_REF], in particular, is to construct quantum GR via a perturbative expansion around the topological theory (see also [START_REF] Witten | 2+1)-Dimensional Gravity As An Exactly Soluble System[END_REF]). This approach is radically new and raises a number of questions. In particular, the unperturbed theory has no local degrees of freedom. Does it make sense to expand a non-trivial QFT around a topological theory? In this paper, we address this question by studying such expansion in a simpler case: Yang-Mills (YM) theory on a flat background.

In fact, YM theory too can be written as a modified BF theory [START_REF] Halpern | Field Strength Formulation Of Quantum Chromodynamics[END_REF], sometimes called BFYM. The conventional perturbative expansion of BFYM, where the zero'th order is the kinetic term, has been investigated in detail in the literature (see for instance the last reference in [START_REF] Halpern | Field Strength Formulation Of Quantum Chromodynamics[END_REF]). Here we show, however, that the coupling constant g 0 can be moved in such a way that the zero'th order is BF theory. Therefore, the expansion in g 0 becomes an expansion around the topological theory, precisely as the one proposed for quantum gravity. We thus investigate the perturbative expansion of the theory in this formulation. In particular, we compute the propagator in a power series in g 0 .

We find that the resulting expansion is equal to the standard one. In the non-abelian case, an interesting subtlety arises: the leading non-vanishing order is abelian, as expected, in spite of the non-abelian nature of the zero'th order BF term. We discuss the implications of this result for gravity in the final section 3.

The possibility of expanding YM around BF opens the way to a homogeneous treatment of quantum gravity and YM theory. A proposal in this direction is under study [START_REF] Freidel | Perturbative expansion for 3d Yang-Mills theory coupled to quantum gravity in the spinfoam formalism[END_REF], and the present work provides support for the viability of the approach.

Expansion around BF of Yang-Mills theory

Consider YM theory with gauge group G, say SU(N ). We call A a µ the YM connection. The index a is in the algebra g of G. The YM curvature is

F a µν = ∂ µ A a ν -∂ ν A a µ + f abc A b µ A c ν
, where f abc are the gauge group structure constants. We use greek letters µ = 0, . . . , 3 for spacetime indices and [µν] = µννµ. For a compact notation, we use the formalism of p-forms, where F = 1 2 F µν dx µ ∧ dx ν = dA + 1 2 [A, A] and introduce the Hodge star * , defined by * F µν = 1 2 ǫ µν ρσ F ρσ . We also introduce a g-valued 2-form B a . Consider the following two actions:

1. the standard second-order YM action

S YM [A a µ ] = 1 4g 0 2 d 4 x F a µν F µν a = - 1 2g 0 2
Tr F ∧ * F ;

2. and the first order action

S BFYM [B a µν , A a ν ] = i Tr B ∧ F - g 0 2 2 Tr B ∧ * B. (2)
The traces Tr are over the algebra indices. The equivalence of the two actions can be easily checked: the equation of motion

0 = δS BFYM δB a µν = i 2 ǫ µνρσ F a ρσ + g 0 2 B µν a implies B a µν = - i 2g 0 2 ǫ µν ρσ F a ρσ . (3)
Plugging ( 3) into (2) and using ǫ µνλτ ǫ µνρσ = -2δ λτ [ρσ] , we obtain [START_REF] Goroff | The Ultraviolet Behavior Of Einstein Gravity[END_REF]. Therefore, the two actions describe the same classical theory.

For g 0 = 0 the action (2) reduces to the action for BF theory. When using the action (1), one usually rescales the connection A → g 0 A. In this way the coupling constant is moved in front of the commutator term of the curvature, F = dA + 1 2 g 0 [A, A]. For g 0 = 0, thus, the scaled YM action reduces to the quadratic term.

To construct the quantum theory, we define the (Euclidean) n-point Green functions

Γ YMa1...an µ1...µn [x 1 . . . x n ; g 0 ] := 1 Z DA A a1 µ1 (x 1 ) . . . A an µn (x n ) e -1 4
Tr (dA+g0[A,A]) 2 -S gf (A) , (4)
for the (rescaled) action (1), and

Γ BFYMa1...an µ1...µn [x 1 . . . x n ; g 0 ] := 1 Z DB DA A a1 µ1 (x 1 ) . . . A an µn (x n ) e -i Tr B∧F + g 0 2 2 Tr B∧ * B-S gf (B,A) , (5)
for the action (2). Here Z is the partition function, namely the functional integral without the field insertions, and S gf (A) and S gf (B, A) are appropriate gauge-fixing terms. In the following we use the Lorentz gauge. Formally, these Green functions are equivalent: integrating over the B field in (5), we indeed obtain

Γ YMa1...an µ1...µn [x 1 . . . x n ; g 0 ] = 1 g 0 n Γ BFYMa1...an µ1...µn [x 1 . . . x n ; g 0 ]. (6)
The factor g 0 -n comes from the rescaling used in (4). Since n-point Green functions are the basic building blocks of QFT, the equivalence above would guarantee that the two actions give rise to the same quantum theory. However, the equivalence is only formal, until we can actually define and evaluate the integrals. In QFT, the conventional way of doing so is by expanding the functional integrals in g 0 around a gaussian integral, that can be evaluated. If the equivalence holds, then the two expansions should coincide, order by order. That is, (getting rid of indices and arguments)

Γ YM(0) + g 0 Γ YM(1) + g 0 2 Γ YM(2) + . . . = 1 g 0 n Γ BFYM(0) + g 0 2 Γ BFYM(1) + g 0 4 Γ BFYM(2) + . . . . (7)
However, there are different reasons for doubting this equivalence, and it is far from obvious how the two expansions could coincide. In particular, the two zero'th orders are extremely different:

• In (4), the zero'th order is a free abelian theory, namely a gaussian integral, which depends on the metric and has lost track of the non-abelian structure,

Γ Ma1 ...an µ1...µn [x 1 . . . x n ] := 1 Z DA A a1 µ1 (x 1 ) . . . A an µn (x n ) e -1 4
Tr (dA) 2 -S gf (A) .

• In ( 5), the other way around, the zero'th order is not gaussian, the theory depends on the non-abelian structure and has lost track of the metric of spacetime,

Γ BFa1...an µ1...µn [x 1 . . . x n ] := 1 Z DB DA A a1 µ1 (x 1 ) . . . A an µn (x n ) e -i Tr B∧F -S gf (B,A) .
How can starting points that are so different give rise to the same Green functions? Below we show that they do. The differences are harmless, and (7) holds, order by order. For simplicity, we focus on the 2-point Green function.

The perturbative expansion of (5) reads

Γ BFYMab µν [x, y; g 0 ] = 1 g 0 2 1 Z ∞ k=0 1 k! ( g 0 2 2 ) k DB DA A a µ (x)A b ν (y) Tr B ∧ * B k e -i Tr B∧F -S gf (B,A) = 1 g 0 2 Γ BFYM(0)ab µν [x, y] + Γ BFYM(1)ab µν [x, y] + g 0 2 Γ BFYM(2)ab µν [x, y] + . . . (8)
For k = 0 we have

Γ BFYM(0)ab µν [x, y] = 1 Z DB DA A a µ (x)A b ν (y) e -i Tr B∧F -S gf (B,A) ; (9)
the gauge-fixing term for the B field is needed to cancel the components of the B fields, which are integrated over, but do not enter the BF action because of the Bianchi identity on the curvature. We perform the DB integral and obtain

Γ BFYM(0)ab µν [x, y] = 1 Z DA A a µ (x)A b ν (y) δ(F ) e -S gf (A) ≡ 0, (10)
because in the Lorentz gauge F = 0 implies A = 0. Therefore, the zero'th order does not contribute to the physics: its correlators vanish.

The leading order of ( 7) is then proved if we can show that, in the Lorentz gauge,

Γ BFYM(1)ab µν [x, y] = Γ Mab µν [x, y] = d 4 p (2π) 4 e -ip (x-y) p 2 η µν - p µ p ν p 2 δ ab . (11)
To do so, we consider separately the abelian and the non-abelian cases.

Free propagator: abelian case

In the abelian case, the curvature is simply given by the exterior derivative of the connection, F = dA, and both the curvature and the B field are gauge-invariant. The gauge-fixing term only concernes the connection. To calculate the propagator, we introduce a source j µ (x) for the connection, and write (j

• A stands for j µ A µ ) Γ BFYM(1) µν [x, y] = 1 2 DB DA A µ (x)A ν (y) B ∧ * B e -i B∧dA-S gf (A) = = - 1 2 δ 2 δj µ (x)δj ν (y) DB DA B ∧ * B e -i B∧dA-S gf (A)+i j•A j=0 . ( 12
)
We perform first the DA integral. Because of the Lorentz gauge-fixing term, this integral selects only the transverse components of the source, j µ T := (δ µ ν -p µ pν p 2 )j ν . Integrating by parts in the action, we thus get

Γ BFYM(1) µν [x, y] = - 1 2 δ 2 δj µ (x)δj ν (y) DB B ∧ * B δ( * dB + j T ). (13)
In order to evaluate the remaining integral, we have to find solutions of the equation * dB = -j T , which in Fourier-transformed components reads 1 2 ǫ µνρσ p ν B ρσ (p) = -j µ T (p). This can be easily solved in B, to give

ǫ αβρσ B ρσ (p) = 2 p 2 p [α j β] T (p).
It follows that

DB B ∧ * B δ( * dB + j T ) = -2 d 4 p (2π) 4 1 p 2 j α T (p) j T α (-p).
Plugging this result in [START_REF] Conrady | Minkowski vacuum in background independent quantum gravity[END_REF] it is straightforward to obtain

Γ BFYM(1) µν [x, y] = d 4 p (2π) 4 e -ip (x-y) p 2 η µν - p µ p ν p 2 .
It is easy to check that in this abelian case Γ BFYM(k) for all k > 1, so that we actually verify (11) exactly.

Free propagator: non-abelian case

In the non-abelian case, the situation is more subtle. As above, we introduce a source j µ a for the connection, but also a source η µν a for the B field. We can then write (here η • B stands for η µν B µν )

Γ BFYM(1)ab µν [x, y] = 1 2 DB DA A a µ (x)A b ν (y) Tr B ∧ * B e -i Tr B∧F -S gf (B,A) = = 1 8 δ 2 δj µ a (x)δj ν b (y) dz δ δη c ρσ (z) δ δη ρσ c (z)
DB DA e -i Tr B∧F -S gf (B,A)+i Tr j•A+i Tr η•B j=η=0 .

Once again, the gauge-fixing term for B is needed to cancel the components which do not enter the BF action. It is convenient this time to perform the DB integral first, obtaining

Γ BFYM(1)ab µν [x, y] = 1 8 δ 2 δj µ a (x)δj ν b (y) dz δ δη c ρσ (z) δ δη ρσ c (z) DA δ * F (A) -η e -S gf (A)+i Tr j•A j=η=0 = 1 8 δ 2 δj µ a (x)δj ν b (y) dz δ δη c ρσ (z) δ δη ρσ c (z) e i Tr j•Aη j=η=0 = = - 1 8 dz δA η a µ (x) δη ρσ c (z) δA η b ν (y) δη c ρσ (z) η=0 . ( 14
)
Here we have introduced the solution A η of the equation

F a µν (A) = ǫ µνρσ η ρσ a (15)
for the connection. The equation is non-linear, and we do not know the analitic form of the solution.

However, as we see from [START_REF] Oeckl | A 'general boundary' formulation for quantum mechanics and quantum gravity[END_REF], only the linear dependence of A η on η matters. Let us therefore expand A η in a power series in η,

A η a µ (x) = dy G 1 ab µνρ (x, y) η νρ b (y) + dydz G 2 abc µνρστ (x, y, z) η νρ b (y)η στ c (z) + . . . (16)
When we insert ( 16) in ( 15), the leading order, in Fourier components, is

p [µ G 1 ab ν]αβ (p) η αβ b (p) = ǫ µνρσ η ρσ a (p).
Remarkably, G 1 is only sensible to the abelian structure of the theory. In the Lorentz gauge we can solve this equation, obtaining

G 1 ab ναβ (p) η αβ b (p) = 1 p 2 ǫ µνρσ p µ η ρσ a (p). (17)
Using this in [START_REF] Oeckl | A 'general boundary' formulation for quantum mechanics and quantum gravity[END_REF] we immediately get

Γ BFYM(1)ab µν [x, y] = - 1 2 dz G 1 ac µρσ (x, z) G 1 bc νρσ (y, z) = = d 4 p (2π) 4 e -ip (x-y) p 2 η µν - p µ p ν p 2 δ ab . (18)
The non-abelian structure, yet present at the beginning of the calculation, plays no role in the final result. Notice how the non-abelian structure comes into play at the next order: when we evaluate Γ BFYM (2) , the term Tr B ∧ * B 2 gives four functional derivatives δ/δη, which make the quadratic dependence of A η on η enter the final expression. But [START_REF] Modesto | Particle scattering in loop quantum gravity[END_REF] at quadratic order in η depends on the non-abelian structure, and we expect to obtain the usual g 0 2 corrections to the free propagator. We do not perform here the explicit calculations.

From the discussion above, we conclude that the perturbative expansion around BF of YM theory on flat spacetime coincides with the conventional one.

Consequences for gravity

Two different formulations of GR as modified BF theories have been considered in the literature, as starting point for the quantisation:

1. The Plebanski action,

S[B IJ µν , ω IJ µ , µ IJKL ] = 1 16πG Tr B ∧ F (ω) + µ IJKL B IJ ∧ B KL , (19)
used in [START_REF] Plebanski | On the separation between Einsteinien substructure[END_REF]. The latin indices are in the group SO(3, 1) (or SO(4) in the Riemannian version). This has been used for the quantum gravity models described in [START_REF] Barrett | Relativistic spin networks and quantum gravity[END_REF].

2. The MacDowell-Mansouri action,

S[B IJ µν , ω IJ µ ] = Tr B ∧ F (ω) -8πG ǫ IJKLM v M B IJ ∧ B KL , (20)
used in [START_REF] Macdowell | Unified Geometric Theory Of Gravity And Supergravity[END_REF][START_REF] Smolin | General relativity with a topological phase: An action principle[END_REF]. The latin indices are in the group SO(4, 1) (SO [START_REF] Rovelli | The Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum gravity model in the loop representation basis[END_REF] in the Riemannian version). This is a more recent approach to the problem, and its quantisation has been considered in [START_REF] Freidel | Quantum gravity in terms of topological observables[END_REF].

Both actions describe modified BF theories. However, there is a substantial difference between the two. The additional term in (19) is a constraint, whereas the additional term in (20) is a genuine interaction. The B field in (19) is a fundamental variable (and µ is a Lagrange multiplier), while the B field in (20) is only a Lagrange multiplier. Notice also the different position of the Newton's constant G.

The perturbative treatment of the additional term appears more natural with the action (20). In the rest of the discussion we focus on this approach. We assume 8πG/ ≪ 1, and we discuss the perturbative expansion

Z GR = ∞ n=0 1 n! ( -i8πG ) n DB Dω ǫ IJKLM v M B IJ ∧ B KL n e i Tr B∧F (ω) (21)
of the partition function of quantum GR.

It is important to distinguish the two cases, of compact or non-compact spacetime. If spacetime is noncompact, appropriate boundary conditions on the fields must be chosen, in order for the functional integrals to be well-defined. In the previous discussion of YM on flat spacetime, we implicitely chose vanishing boundary conditions for the fields at spacial infinity. Once the boundary conditions are chosen, the classical equations of motion select only one solution (up to gauge), and the functional integral defining the Green functions is generically picked around this solution, as discussed above. In this situation, we showed in the previous section that the perturbative expansion around BF coincides with the conventional one. If we treat the non-compact case in gravity, we have to provide spacial boundary conditions for the gravitational field. The natural choice here is asymptotic flatness: g µν → η µν at infinity. When using the action (20), this is a condition on the B field. Once this condition is fixed, we expect the quantum theory to be picked around a classical solution (see the discussion in [START_REF] Mattei | From 3-geometry transition amplitudes to graviton states[END_REF]). In this setting, we expect our analogy with YM theory to hold through: namely, (21) should give rise to the conventional (non-renormalizable) perturbation theory in terms of gravitons. From this perspective, the "expansion around the topological sector" of the theory does not appear to be different from the conventional expansion.

The situation is far less clear in the compact case, where the background field that minimizes the gaussian action, or is selected by the delta function, is not determined by asymptotic conditions. In particular, it has been recently argued that the definition of the theory on a compact spacetime region with boundaries is more suitable to extract physical information from background independent theories [START_REF] Rovelli | Quantum Gravity[END_REF][START_REF] Conrady | Minkowski vacuum in background independent quantum gravity[END_REF][START_REF] Oeckl | A 'general boundary' formulation for quantum mechanics and quantum gravity[END_REF][START_REF] Modesto | Particle scattering in loop quantum gravity[END_REF]. This is perhaps the situation where the proposal of [START_REF] Freidel | Quantum gravity in terms of topological observables[END_REF] appears to be more interesting. We leave the discussion on this case open for further developments.


Javier Borge-Holthoefer
email: borge.holthoefer@gmail.com

Sandro Meloni

Bruno Gonçalves

Yamir Moreno

Keywords: Rumor spreading, Online social networks, Human activity patterns

The burst in the use of online social networks over the last decade has provided evidence that current rumor spreading models miss some fundamental ingredients in order to reproduce how information is disseminated. In particular, recent literature has revealed that these models fail to reproduce the fact that some nodes in a network have an influential role when it comes to spread a piece of information. In this work, we introduce two mechanisms with the aim of filling the gap between theoretical and experimental results. The first model introduces the assumption that spreaders are not always active whereas the second model considers the possibility that an ignorant is not interested in spreading the rumor. In both cases, results from numerical simulations show a higher adhesion to real data than classical rumor spreading models. Our results shed some light on the mechanisms underlying the

Introduction

Understanding the way in which a disease or a piece of information spreads from person to person is of obvious practical relevance. If we are able to comprehend the mechanisms that dominate such spreading processes we would be able to enhance the spread of valuable information through a community or impair an outbreak of an infectious disease. The similitudes between these two processes, epidemics and information diffusion (also referred to as rumor spreading [START_REF] Goffman | Generalization of Epidemic Theory -An Application To the Transmission of Ideas[END_REF][START_REF] Dietz | Epidemics and Rumours: A Survey[END_REF][START_REF] Galam | Modeling rumors: The no plane pentagon french hoax case[END_REF][START_REF] Kimmel | Rumors and rumor control[END_REF][START_REF] Kosfeld | Rumours and markets[END_REF]) have long been recognized and the two fields have evolved in parallel freely borrowing ideas and concepts from each other [START_REF] Daley | Stochastic rumours[END_REF][START_REF] Daley | Epidemic Modelling[END_REF].

In epidemics it has become clear that some individuals that are "superspreaders" play a dominant role in the course of an epidemic [START_REF] Lloyd-Smith | Superspreading and the effect of individual variation on disease emergence[END_REF]. Intuitively, one expects that similarly influential individuals would also be present in the case of information diffusion and recent years have witnessed a growing interest to understand how to identify them [START_REF] Boccaletti | Complex networks: structure and dynamics[END_REF][START_REF] Castellano | Statistical physics of social dynamics[END_REF][START_REF] Watts | Influentials, networks, and public opinion formation[END_REF][START_REF] Rogers | Diffusion of Inovations[END_REF][START_REF] Bakshy | The role of social networks in information diffusion[END_REF]. Successful approaches have focused on studying the effect that different network-based centrality measures have on rumor spreading. In particular, one recent seminal approach [START_REF] Kitsak | Identification of influential spreaders in complex networks[END_REF] has identified the k-core as the best measure to predict influence, outperforming degree centrality or betweenness in the context of an epidemic spreading process. This insight has been followed by many other works, which mainly discuss under which circumstances the k-core actually predicts a node's disease spreading capabilities [START_REF] Castellano | Competing activation mechanisms in epidemics on networks[END_REF] or propose alternative measures of influence [START_REF] Cosley | Sequential influence models in social networks[END_REF][START_REF] Klemm | A measure of individual role in collective dynamics: spreading at criticality[END_REF].

Following the original proposal of Kitsak et al [START_REF] Kitsak | Identification of influential spreaders in complex networks[END_REF], Borge-Holthoefer and Moreno [START_REF] Borge-Holthoefer | Absence of influential spreaders in rumor dynamics[END_REF] studied rumor spreading dynamics to learn whether the k-core could predict authority or not. Surprisingly, their results indicate that a rumor's successmeasured as the number of individuals that learn about the rumor at the end of the spreading dynamicsis topology-independent: no matter who in a network triggers the rumor, the final number of nodes who learn about it will be the same (given the same spreading parameters). Additionally, central nodes (those at the highest core levels) behave as firewalls, short-circuiting the capacity of the rumor to spread further. This theoretical prediction is clearly at odds with empirical evidence and points to a shortcoming of theoretical models that must be overcome.

The development of the Web 2.0 and the growing popularity of online social networks have not only had a tremendous impact on our daily lives, but they also had the beneficial consequence of generating detailed data on social communication patterns, which can ultimately inspire and guide the development of more realistic models. In this paper we try to fill the gap between observations from real systems and theoretical predictions by introducing some simple modifications to models proposed previously [START_REF] Moreno | Efficiency and reliability of epidemic data dissemination in complex networks[END_REF][START_REF] Nekovee | Theory of rumour spreading in complex social networks[END_REF]. The resulting models are able to better approximate the behavior of users as observed in online social networks, in particular, the fact that there are influential nodes with larger diffusion capacities, an important feature not accounted for with current rumor spreading models.

Our analysis starts from the simple empirical observation [START_REF] Barabasi | The origin of bursts and heavy tails in human dynamics[END_REF][START_REF] Perra | Activity driven modeling of dynamic networks[END_REF][START_REF] Borge-Holthoefer | Locating privileged spreaders on an online social network[END_REF]] that individuals display complex activity patterns both on and offline and, in particular, are not active around the clock. This fact has two possible interpretations. On one hand, users that are actually spreading the rumor are active only at specific times and only then they are able to participate in the diffusion process. On the other hand, an individual's choice of becoming active and participating in a specific information cascade can be seen as a demonstration of interest in the topic and his/her will to spread it. Inspired by these two interpretations, we derive two different rumor diffusion models.

The first model incorporates the differences in the activity of the individuals responsible for the spreading of the rumor. Each spreader is assigned with a randomly chosen probability of being active at a given time. In this context, we study the effects of the heterogeneity [START_REF] Perra | Activity driven modeling of dynamic networks[END_REF] in the activation probability extracting values from three different probability distributions ranging from a uniform to a long-tailed one. In a more realistic version, following the idea that more active users, usually, also have a central role in the topology of the network, we relate the activity of each individual with its degree.

The second model takes into account the fact that an individual could learn the rumor without actually spreading it further. This is for example what happens in most online social networks, in which followers receive pieces of information from those they are following and not alwaysindeed, rarely they transmit the news further. We therefore introduce the possibility that a person that comes into contact with the rumor does not spread it anymore. This approach is complementary to the previous one, as we consider that it is the ignorants and not the spreaders who can be inactive.

In the remaining of the paper, we will show that even though these alternatives introduce only small and intuitive changes, they are able to shed light on the complex social mechanisms at work in real social systems and, at least qualitatively, reproduce the heterogeneities observed in Twitter data. The rest of the manuscript is organized as follows: In the next section, we present a general framework for rumor spreading on networks while subsections 2.1 and 2.2 present the two modified models and the results of numerical simulations. Finally, we draw our conclusions in section 3.

General modeling framework

In classical rumor spreading models on networks, each of the N nodes of a network can be in one of three possible states. A node holding a rumor and willing to transmit it is called a spreader. Nodes that are unaware of the update will be referred to as ignorants, while those that already know it but are not willing to spread it further are called stiflers. We denote the density of ignorants, spreaders, and stiflers at time t as i (t), ρ (t) and r (t), respectively, with i (t) + ρ (t) + r (t) = 1, ∀t. The spreading process takes place along the Fig. 1 (color online) Density of stiflers at the end of the rumor spreading r∞(kS) originated in a node that is part of the kS k-core for the general model of rumor spreading (squares) and the empirical fraction, Nc/N , of users reached by cascades originated at nodes with k-core kS (circles) as extracted by real Twitter data (for details see [START_REF] Borge-Holthoefer | Absence of influential spreaders in rumor dynamics[END_REF][START_REF] Gonzalez-Bailon | The dynamics of protest recruitment through an online network[END_REF]). Numerical simulations were ran using the same empirical Twitter Follower network.

links connecting spreaders and ignorants. At each time step, spreaders contact all of their neighboring nodes. In the simplest case, whenever a spreader j contacts a node n that is ignorant, the latter will become a spreader with a fixed probability λ. Otherwise, if n is already a spreader, the node j will turn into a stifler with probability α. Mathematically, the general model can be represented as:

I λ -→ S S α -→ R
where the initial conditions are set such that i (0) = 1 -1/N , ρ (0) = 1/N and r (0) = 0. In addition, and without loss of generality, we set λ = 1 unless other values are explicitly stated.

For each alternative model presented, extensive numerical calculations have been carried out by simulating the dynamics of rumor propagation on top of a real-world Twitter following/follower network [START_REF] Gonzalez-Bailon | The dynamics of protest recruitment through an online network[END_REF]. From an initial scenario, in which all nodes belong to the ignorants class except the seed, we perform S = 10 simulations. This is repeated for each node, i.e. every vertex of a network of N nodes acts as the initial seed S times, to obtain statistically significant results. In this way, for each node i, we average the final density of stiflers in the network r i ∞ . This quantity accounts for the spreading capacity of node i, which quantifies how deep the rumor penetrated the network when node i was the initial seed:

r i ∞ = 1 S S m=1 r i,m ∞ (1)
where r i,m ∞ represents the final density of stiflers for a particular run m with origin at node i. With this information at hand for all nodes, we coarse-grain the individual r i ∞ 's into classes of nodes according to their core number. Thus, r ∞ (k S ) represents the average stifler density for all runs with a seed with a k S core index:

r ∞ (k S ) = i∈Υ k S r i ∞ N kS (2)
where Υ kS is the set of all N kS nodes with k S values.

Figure 1 shows the comparison between the values of r ∞ (k S ) as obtained via the numerical simulations of the above rumor spreading model and the observed fraction of users N c /N reached by cascades originated at nodes with core index k S obtained by analyzing Twitter usage data from the Spanish Indignados movement (see [START_REF] Borge-Holthoefer | Absence of influential spreaders in rumor dynamics[END_REF][START_REF] Gonzalez-Bailon | The dynamics of protest recruitment through an online network[END_REF] for details on how data have been extracted and analyzed). The differences observed in this plot are striking. Even though the model ran on the exact same network, the theoretical prediction is completely insensitive to the value of the originating k-core, while in the empirical data there is a clear correlation between belonging to higher cores and larger numbers of nodes reached by the cascade. This difference in behavior clearly shows that there is something fundamentally lacking in the theoretical model.

Model I: Human activity and temporal patterns

We next consider the possibility that nodes are not always available to take part in a certain communication exchange. Each individual is active with a certain probability, a i , affecting his/her behavior as a spreader. Thus, on top of the constraints of the basic framework presented above, we assume that a spreader only attempts to spread the rumor when it is active. As a consequence, the transition from the class of ignorants to the class of spreaders happens less often.

It is worth mentioning that as far as our model is concerned, the approach adopted is rooted in the observation that human activity patterns are mostly heterogeneous and therefore individuals are not always active [START_REF] Perra | Activity driven modeling of dynamic networks[END_REF] nor is their activity distributed randomly over time [START_REF] Barabasi | The origin of bursts and heavy tails in human dynamics[END_REF][START_REF] Gonçalves | Human dynamics revealed through web analytics[END_REF][START_REF] Vazquez | Impact of non-poissonian activity patterns on spreading processes[END_REF]. However, we assume that nodes in the network still have memory of who their potential neighbors are, and although not all the links of a given node were concurrently active, the set of available neighbors would be predefined by the underlying static (aggregated) topology. A more accurate description would require to consider that the topology is shaped by the activity of the nodes, so that the resulting time-varying networks are activity-driven [START_REF] Perra | Activity driven modeling of dynamic networks[END_REF]. In the latter case, the interactions between the different classes of nodes in the system would still be activity-driven, but no memory of the static topology would be present, as the interaction structure is redefined at each time step. Whether or not both mechanisms lead to similar behavior is a matter that deserves further investigation.

On the other hand, note that being active or not has no effect on the rumor's recipients (ignorants). This mechanism is specific for asynchronous communication systems such as Twitter, FedEx, email or SMS where information can be sent even without requiring the collaboration of the recipient. On the contrary, for synchronous systems, such as phone calls or Instant Messaging, that require both the source and the target of a message to be active at the same time, such a scheme would not suffice.

Here we explore three possibilities for the activity distribution: (a) uniform, P (a) ∼ c; (b) exponential, P (a) ∼ e -a/ac ; and (c) power-law, P (a) ∼ a -γ . Interestingly, these distributions yield completely different results. Figure 2 illustrates this perfectly. The increase of heterogeneity in activity patterns moves the distribution of outbreak sizes, r ∞ (k S ), closer to empirical results, highlighting the fact that heterogeneity is a fundamental factor in real information spreading processes. A uniform activity distribution (lowest panel) completely flattens the spreading capabilities, no matter if nodes are in a topologically relevant region or not. This is in good agreement with [START_REF] Borge-Holthoefer | Absence of influential spreaders in rumor dynamics[END_REF], the only difference being the time the system needs to reach a final state (that is, the probabilities delay the process significantly). An exponential distribution introduces some amount of asymmetries in the activity distribution, which slightly affects the spreading results (central panel). Finally, a power-law probability distribution introduces heterogeneity in the spreading success, the higher k S , the higher the spreading capacity, just as it has been found empirically [START_REF] Borge-Holthoefer | Locating privileged spreaders on an online social network[END_REF][START_REF] Gonzalez-Bailon | The dynamics of protest recruitment through an online network[END_REF].

The importance of the spreader-to-stifler rate is revealed in the heterogeneous scenario. α sets the timescale relevant to this process. For high values of α, ρ nodes quickly become stiflers and the rumor doesn't have the possibility of reaching a significant fraction of the population while lower values of α easily allow for successful dissemination. Furthermore, it should be noted that we are assigning activity levels entirely at random, without any relation between topological features and activity probabilities. This means that a poorly connected node is just as likely to be highly active as a node with high degree. However, it has been seen previously [START_REF] Perra | Activity driven modeling of dynamic networks[END_REF] that activity distributions are correlated with the observed degree distribution. The simplest form of effectively implementing this correlation is to assign to node i an activity probability a i = k i /k max .

Figure 3 illustrates the results of this scenario. The great heterogeneity of the degree distribution is clearly reflected. Rumors triggered from low degree nodes (which necessarily have low k S ) die out soon, because the nodes they reach are almost never active. On the contrary, high degree nodes (which are more likely to belong to a high k core) persistently forward messages, turning any rumor into system-wide knowledge. Note that spreading is almost identical regardless α, in stark contrast to the upper panel of Figure 2 (where α determines the shape of spreading) possibly indicating that higher level correlations also play an important role.

Figure 4 shows the comparison between the topology-dependent and random distribution of the activation probabilities. In this case, all the curves have been obtained with a power-law activity distribution. However, in one of them (blue diamonds) the activity of each node is proportional to its degree (a i = k i /k max ), whereas in the other curves activation probabilities are assigned at random and thus independently from the topological features of the nodes. As in Fig. 2 in the randomly distributed case the spreading is highly affected by α meanwhile for the degree-dependent activation probabilities a substantial independence from α is present.

Model II: Apathy

The analysis of real data from online social networks demonstrates that most of the time users do not react to received messages [START_REF] Borge-Holthoefer | Locating privileged spreaders on an online social network[END_REF]. One possible interpretation for this is that they have been informed of a rumor but chose not to spread it. This interpretation suggests another ingredient that might be missing from classical rumor spreading models and that might help bring them closer to reality: the possibility that an ignorant is apathetic and directly Fig. 3 (color online) Fraction of stiflers at the end of the rumor spreading r∞(kS) originated in a node of k-core kS when activation probabilities are proportional to nodes degree. Three different values of α are used, the underlying network is the empirical Twitter follower network from Ref. [START_REF] Gonzalez-Bailon | The dynamics of protest recruitment through an online network[END_REF].

goes to the stifler status and does not participate further in the spreading dynamics. As noted before, this kind of behavior is common in online social networks like Twitter, in which one receives messages that are rarely spread further. We incorporate this new element by introducing the probability, p, that an ignorant is interested in the topic and decides to diffuse it. In this scenario, when a spreader contacts an ignorant, the latter turns into a spreader with probability λp and into a stifler with probability (1 -p) λ.

The transitions allowed by our model are then:

I λp -→ S S α -→ R I λ(1-p) -→ R
It should be noted that Model II is a natural counterpart of Model I presented in section 2.1, in the sense that it also assigns activity probabilities to each node. The main difference is that this uniform probability p is assigned to ignorant individuals and determines whether or not they choose to participate in the spreading process. A parallel can also be made to the case of epidemic spreading where a person become immune to a disease upon Fig. 4 (color online) Fraction of stiflers at the end of the rumor spreading r∞(kS) originated in a node of k-core kS when the activation probability is proportional to nodes degree (diamonds) or randomly distributed (squares, triangles and circles).

Network topology and the other parameters are the same as in Fig. 3.

coming in contact with pathogen and before it is able to develop symptoms or spread it further.

The behavior of the system can be better understood analytically by writing the mean-field rate equations governing its time evolution in the homogeneous mixing approximation:

∂i (t) ∂t = -λp kρ (t) i (t) -(1 -p) λ kρ (t) i (t) , (3)
∂ρ (t) ∂t = λp kρ (t) i (t) -α kρ (t) (ρ (t) + r (t)) , (4)
∂r (t) ∂t = α kρ (t) (ρ (t) + r (t)) + (1 -p) λ kρ (t) i (t) , (5)
with the initial conditions i (0) = 1 -1/N, ρ (0) = 1/N , r (0) = 0 and where k represents the number of contacts each spreader has per unit time. The first term in the right side of Eq. 3 accounts for the density of ignorants that turn into spreaders after an interaction whereas the second term model the ignorant to stifler transition with probability (1 -p) λ. Recalling that i (t) + ρ (t) + r (t) = 1 we can study the system of Eqs. 3-5 analytically in the infinite-time limit ρ(∞) = 0, obtaining:

r ∞ = 1 -e -(1+ λ α p)r∞ . (6)
The average total stifler density r ∞ , for various values of p, obtained by numerically solving this transcendental equation is shown in Figure 5. We also performed a series of Monte-Carlo (MC) simulations in the homogenous mixing limit. At t = 0 the entire population is ignorant with only a small fraction (≃ 1/N ) being spreaders. At each time step, each spreader contacts k individuals chosen at random from the entire population. If the chosen individual is an ignorant it will become a spreader with probability λp or directly move to stifler status with (1 -p) λ. Otherwise, when a spreader comes in contact with a stifler or another spreader it turns into a stifler with probability α. When the spreading process reaches the absorbing state ρ (t) = 0 the final density of stiflers is recorded. The simulation results are also plotted in Figure 5 for comparison with the analytical solution. The agreement between the two approaches is striking and serves as a confirmation that we are not missing any fundamental ingredients in our analyses.

Although the addition of a constant p parameter (any node is assigned the same p) is a crude approximation to the interest that ignorants might have in becoming spreaders, it has profound implications for the system dynamics when compared to the standard setup. Figure 6 shows the behavior of the system with the inclusion of the new rule, with fixed λ = 1, α = 0.5 for different p values. As for the power-law activity distribution in model I and in real data [START_REF] Borge-Holthoefer | Locating privileged spreaders on an online social network[END_REF][START_REF] Gonzalez-Bailon | The dynamics of protest recruitment through an online network[END_REF][START_REF] Perra | Activity driven modeling of dynamic networks[END_REF] a strong correlation between the k-core of the seed and the final outcome of the spreading is observed. In particular, and although we have made no efforts toward fitting this value, it is clear that for a very low probability (p = 10 -3 ) we already have a close-to-real behavior. Although this value might seem low (only one in a thousand contacted individuals do forward the rumor), one must consider that this is the probability that one individual will choose to participate in any of rumors he observes. It is well known that most Twitter users commonly follow on the order of hundreds of other individuals so that the number of pieces of content they are exposed to daily can easily be on the order of thousands or tens of thousands of which they are only able, or willing, to participate in a few.

Conclusions

Online social networks are becoming increasingly central in our lives as they come to permeate our daily activity. It then comes as no surprise that they have been welcome by mass social movements around the world as unique platforms for the diffusion of new ideas and even for the coordination of large numbers of individuals. Understanding the forces that drive the behavior of individuals interacting in these networks is then one of the great challenges for science in the next years. One interesting aspect is how ideas are shared between individuals and what are the conditions that allow for a large dissemination of them. In this context several works studied how a rumor can spread in a population of ignorant individuals but, due to the changes in the way in which these tools allow us to communicate, most of those works cannot catch the details of rumor dynamics on such large scale social systems.

Driven by data from a microblogging online platform we propose two modifications to classical rumor spreading models that are able to qualitatively reproduce the observed differences in the number of individuals reached by the rumor when the seed is located in the most connected circles of the network or in its periphery. The models we present are based on the observation that individuals, both spreaders and ignorants, are not always active in the network. Each model then implements a different effective mechanism that is consistent with this fact: Model I assigns activity probabilities to each node and allows for spreading to occur only when a spreader node is active while Model II assumes that each node has a finite probability of being interested in spreading each specific rumor and would otherwise chose not to participate in the diffusion process. Both variations have proved effective in bringing the classical model one step closer to reality.

In the case of Model I, numerical results highlight that the more heterogeneous the patterns of activation are the more faithfully we are able to imitate real data. Moreover, if, in a second approximation, we relate the activity of a node with its degree (as higher degrees are commonly correlated with high levels of activity in the network) we also observe a substantial independence of the results from stifler transition ratio, α. For Model II, we were also able to give an analytical expression for the final density of stiflers in the system. Interestingly, the analysis of the numerical simulations suggests that close-to-real results are obtained when the probability for an ignorant to be interested in the rumor is very low; another feature also observed in real social networks.

The results presented in this paper clearly evidence that classical rumor spreading models are severely short on their ability to effectively approximate reality. We have shown that even small, empirically based, modifications can significantly increase their level of realism. In particular, our results shed some light on the interplay between technology and human interactions that are at the origin of some of the complex behaviors we observe daily. With this work we have taken a significant first step in paving the way toward a deeper understanding of how ideas spread through our online and offline social networks and help shape current events and society as a whole.

  data r ∞ (k S ), λ = 1.0 α = 1.0

Fig. 2 (

 2 Fig. 2 (color online) Density of stiflers at the end of the rumor spreading r∞(kS) originated in a node of k-core kS in the activation probability model. Three different probability distribution functions are used: a power-law with exponent γ = 2.5 (upper panel), exponential with ac = 0.1 (middle panel) and a uniform distribution (lower panel).

  distributed α = 0.5, randomly distributed α = 1.0, randomly distributed a i ∈ (0,1] topology-dependent

Fig. 5 (

 5 Fig. 5 (color online) Fraction of stiflers at the end of the rumor spreading r∞ for different values of p in comparison with the theoretical prediction of Eq. 6. Numerical results are the average over 10 3 stochastic runs. In both cases λ = 1.0, α = 0.5 and k = 4.

Fig. 6 (

 6 Fig.6(color online) Fraction of stiflers at the end of the rumor spreading r∞(kS) originated in a node of k-core kS in the ignorant-to-stifler transition model for different p values. For each value of p both the average and the maximum value of numerical simulations is presented. Network topology and the other parameters are the same as in Fig.3.

Acknowledgements The authors would like to thank A. Rivero for the help in collecting Twitter data and useful discussions. This work has been partially supported by MICINN through Grants No. FIS2008-01240, FIS2009-13364-C02-01 and No. FIS2011-25167, and by the Government of Aragón (DGA) through a grant to FENOL group.