Cooperative EpidemicsSummary of the meeting of Dec. 22 2016We have found numerically that on ER networks, increasing thelevel of clustering leads from a hybrid transition (zero or small clustering) to a continuous transition (large clustering). The line separating the two behavior grows as a function of q: more clustering is needed to see a continuous transition if q is large.For SF (Barabasi-Albert) networks instead, we always see a continuous transition (in agreement with Grassberger's Nature Physics). It is not clear why we see this phenomenology.The argument by Grassberger et al. is that the hybrid transition is caused by few short loops and many long loops. Barabasi-Albert networks have few short loops (although more than ER)and many long loops (as ER). Why then a hybrid transition is not observed on SF networks?One possibility is that the continuous transition we see for SF is only a finite size effect. The effective clustering for the sizeswe can simulate is still too large (it goes to zero only in the limitN \to \infty). It is not possible however to simulate larger systems.To try to understand we will consider the Configuration Model (CM)which interpolates between BA (gamma=3) and ER (gamma=\infty)without any clustering.If we consider values of $\gamma$ equal to 4 or 6 can we see a hybridtransition?Another test we will perform is the analysis of what happens whenthe two diseases are seeded in different nodes randomly chosenin the network.We will consider both ER and BA networks without clustering.What happens to the transition in this case? Is it continuous ordiscontinuous?This test should allow to decouple the effect of the topology on small scale (presence of short loops) from that of the topology at large scale (long loops). The presence of short loops prevents the appearence of a discontinuous transition only because theepidemics cannot spread separately when they start from the same seed. This problem becomes irrelevant when they start from two seeds far apart. If we start the two epidemics in two distinct seeds, any difference in the phenomenology eventually observed on the ER and on the SFmust be ascribed to some difference in the distribution of large loops.Finally, we have realized that the "message passing" approaches typicallyused in ordinary percolation (and SIR) would not give interestingresults for this problem (even if we would be able to formulate themproperly), because they neglect long loops which are instead crucialfor this problem.

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Cooperative Epidemics Summary of the meeting of Dec. 22 2016 We have found numerically that on ER networks, increasing the level of clustering leads from a hybrid transition (zero or small clustering) to a continuous transition (large clustering). The line separating the two behavior grows as a function of q: more clustering is needed to see a continuous transition if q is large. For SF (Barabasi-Albert) networks instead, we always see a continuous transition (in agreement with Grassberger's Nature Physics). It is not clear why we see this phenomenology. The argument by Grassberger et al. is that the hybrid transition is caused by few short loops and many long loops. Barabasi-Albert networks have few short loops (although more than ER) and many long loops (as ER). Why then a hybrid transition is not observed on SF networks? One possibility is that the continuous transition we see for SF is only a finite size effect. The effective clustering for the sizes we can simulate is still too large (it goes to zero only in the limit N \to \infty). It is not possible however to simulate larger systems. To try to understand we will consider the Configuration Model (CM) which interpolates between BA (gamma=3) and ER (gamma=\infty) without any clustering. If we consider values of $\gamma$ equal to 4 or 6 can we see a hybrid transition? Another test we will perform is the analysis of what happens when the two diseases are seeded in different nodes randomly chosen in the network. We will consider both ER and BA networks without clustering. What happens to the transition in this case? Is it continuous or discontinuous? This test should allow to decouple the effect of the topology on small scale (presence of short loops) from that of the topology at large scale (long loops). The presence of short loops prevents the appearence of a discontinuous transition only because the epidemics cannot spread separately when they start from the same seed. This problem becomes irrelevant when they start from two seeds far apart. If we start the two epidemics in two distinct seeds, any difference in the phenomenology eventually observed on the ER and on the SF must be ascribed to some difference in the distribution of large loops. Finally, we have realized that the "message passing" approaches typically used in ordinary percolation (and SIR) would not give interesting results for this problem (even if we would be able to formulate them properly), because they neglect long loops which are instead crucial for this problem.

francesca colaiori