## 1 Introduction  
It may be redundant to say that a correct characterization of mass transport phenomena occurring in particle assemblies, membranes, and porous media is crucial in a number of applications in engineering, chemistry, and physics. The fluid path through a catalyst and/or adsorbent particle beds can severely affect the performance of a mass transfer-based process, and a number of works of literature are aimed at optimising the catalyst distribution [1; 2; 3; 4; 5]. However, for this purpose, it is required to understand not only the dependence of the transport properties on the operating conditions (temperature, pressure, etc.) but also the morphology of the structure, which are characteristics involving several important geometrical parameters, like specific area, porosity, tortuosity, etc. Among such parameters, relatively minor attention is paid to the so-called connectivity of a structure, representing a property measuring a sort of degree of connection of the inner voids. However, if the meaning of connectivity is relatively intuitive, its formal definition and quantification are not obvious at all. For example, [PERSON] et al. (2015) provide a sort of quantification of connectivity by reconstructing the considered structures through image analysis techniques, classifying all the inner void channels into connected, disconnected, and unknown voids, and simply evaluating the percentage of the connected void volume to the total volume [6]. Slightly differently, [PERSON] et al. (2009) defined a connectivity factor in terms of percentage of voxels connected in all directions [7]. However, the drawback of these approaches is that they cannot be applied to all the cases in which there are experimental evaluations of tortuosity [8; 9; 10; 11; 12; 13], where there are no voxels that it is possible to analyse directly. In informatics, connectivity of a communication network is generally related to the ease for nodes of exchanging information pieces, which are transported along available branches (i.e., connection paths). By analogy, the connectivity of a physical structure can be defined as the ease for a particular phenomenon of propagating within a particular mean. From this point of view, the key-concept to understand in defining a proper connectivity factor for a material is that, in principle, connectivity depends on the particular transport phenomenon involved in the transfer. This can be understood if considering that, for example, two generic internal points inside a porous structure belonging to solid and void volume, respectively, are disconnected with respect to diffusion but connected with respect to heat transfer. In this work, we are interested in quantifying the connectivity that measures the degree of physically connected voids, where at least one path exists along which a hypothetical molecule can reach any two points of the void space by diffusion without interpenetrating into the solid space (allowed path, Figure 1a). We can refer to this type of connectivity as diffusional connectivity, whose mathematical definition must depend on geometry only to be effective for a morphological characterization. For this purpose, we choose and propose here to define a Connectivity Factor (\(\varphi\)) as the inverse of the diffusional tortuosity \(\tau\) (Equation (1)):  
\[ \begin{array}{l} Connectivity\ Factor\ \varphi\equiv\frac{1}{\mathit{Tortuosity}\ \tau}\\ \tau\rightarrow\infty\ \Leftrightarrow\ \varphi\to 0\\ \tau=1\ \ \Leftrightarrow\ \varphi=1\end{array}  \tag{1}\]  
It is highlighted that, according to Equation (1), the connectivity factor \(\varphi\) is comprised within [0, 1], this giving an immediate idea of the connectivity degree of the considered structure.